Multiple Dirichlet Series and Shifted Convolutions
MMULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS
JEFF HOFFSTEIN AND THOMAS A. HULSE
Dedicated to Winnie Li with admiration.
Abstract.
We define, and obtain the meromorphic continuation of, shifted Rankin-Selberg convolutions in one and two variables. As sample applications, this continuationis used to obtain estimates for single and double shifted sums and a Burgess-type boundfor L -series associated to modular forms of arbitrary central character. Further applica-tions are furnished by subsequent works by the authors and their colleagues. Introduction
In 1939 and 1940 in [42] and [46], Rankin and Selberg independently introduced what isnow called a Rankin-Selberg convolution. Very roughly, if L ( s, f ) = (cid:88) m ≥ a ( m ) m s + k − and L ( s, g ) = (cid:88) m ≥ b ( m ) m s + k − are two L -series corresponding to modular forms, each convergent when Re s >
1, witha functional equation in s , and a meromorphic continuation to C , the Rankin-Selbergconvolution provides a functional equation and a meromorphic continuation to C of thenew L -series L ( s, f ⊗ g ) = (cid:88) m ≥ a ( m )¯ b ( m ) m s + k − . This idea had a profound impact on the field and was responsible for what was, at thattime, the best progress toward the Ramanujan conjecture.In 1965, in [47], a paper summing up progress in modular forms, Selberg introduced thenotion of a shifted convolution. If h is a positive integer, his shifted convolution was theseries (cid:88) m ≥ a ( m )¯ b ( m + h )(2 m + h ) s + k − . (1.1)This series converges locally normally for Re s > Date : September 18, 2018. a r X i v : . [ m a t h . N T ] N ov JEFF HOFFSTEIN AND THOMAS A. HULSE
Since then it has been generally recognized that the analytic properties of series such as(1.1) play a very important role in estimating the size of certain shifted sums, such as (cid:88) m The subconvexity problem for GL (2) can be posed inseveral different aspects and here we will focus on the conductor of a twisting character inthe context of holomorphic cusp forms. Specifically, let f be a primitive holomorphic cuspform of even weight k and character χ for Γ ( N ), with N square free, having Fourierexpansion f ( z ) = (cid:88) m ≥ A ( m ) m ( k − / e πimz . For χ a Dirichlet character mod Q , we write the twisted L -series as L ( s, f, χ ) = (cid:88) m ≥ A ( m ) χ ( m ) m s . ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 3 This possesses a functional equation as s → − s , and the convexity bound for the centralvalue which follows is L ( , f, χ ) (cid:28) f Q + (cid:15) . The Lindel¨of Hypothesis in the Q aspect states that the exponent can be reduced from + (cid:15) to (cid:15) , and any bound that reduces to − δ , for some δ > GL (1) is 1 / L -functions [12, 13, 14]. Anothercelebrated result, in the context of GL (1) and a Dirichlet L -series, was obtained by Conreyand Iwaniec, [10], who showed that L ( , χ ) (cid:28) Q / (cid:15) , where the character χ is real and primitive with odd conductor Q . This is analogous tothe Weyl bound for the Riemann zeta function, and it remains the best known result fora GL (1) L -series where Q is varied over odd integers.Progress has been made in GL (2) by a number of different authors, with the most re-cent and strongest results in the twisting parameter being obtained by Blomer and Har-cos. In [7], they demonstrate that for f a primitive holomorphic or Maass cusp form ofarchimedean parameter µ , level N and trivial nebentypus, L ( , f, χ ) (cid:28) (cid:15) (cid:16) | µ | N / q / + | µ | N ( N, q ) / q / (cid:17) ( | µ | N q ) (cid:15) when f is holomorphic, and L ( , f, χ ) (cid:28) (cid:15) (cid:16) (1 + | µ | ) N / q / + (1 + | µ | ) / N ( N, q ) / q / (cid:17) (1 + | µ | ) N q ) (cid:15) in the Maass form case.In [8], Blomer and Harcos work over an arbitrary totally real field K . In this contextthey let π be an irreducible cuspidal representation of GL (2 , K ) \ GL (2 , A K ), with unitarycentral character and χ a Hecke character with conductor of norm Q . They also let θ ,with 0 ≤ θ ≤ , be the best progress toward the Ramanujan-Petersson conjecture. (Here θ = 7 / 64 is the best known bound. See [31] for Q and [6] for an arbitrary number field.The Ramanujan-Petersson conjecture predicts θ = 0.) They obtain the bound L ( , π ⊗ χ ) (cid:28) π,χ ∞ ,K,(cid:15) Q − (1 − θ )+ (cid:15) . Here, as our point is to demonstrate a proof of concept for a new method, we work in amore restricted setting than in the work of Blomer and Harcos, restricting ourselves to Q , and the case of f holomorphic of square free level with even weight. In Section 9 weprove the following Theorem 1.1. Let f be a primitive holomorphic cusp form of even weight k , character χ and fixed square free level N , and let χ be a Dirichlet character modulo Q . Then L ( , f, χ ) (cid:28) f,(cid:15) Q / θ/ (cid:15) . JEFF HOFFSTEIN AND THOMAS A. HULSE Remark 1.2. Since an earlier version of this paper was circulated, Blomer and Harcoshave succeeded in removing the dependence on nebentypus and the Ramanujan conjec-tures in an addendum to their paper [7]. We, in the opposite direction, have discoveredthat we are unable to remove it. Thus we note that earlier versions of this paper containthe mistaken claim that the bound we are able to obtain is independent of progress towardthe Ramanujan Conjecture.The techniques of this paper have been further generalized to produce other subconvexityresults. In [34], Kuan obtained a hybrid bound for L ( + it, f, χ ) in the t and Q -aspects.In [32], Kıral was able to produce a completely analogous bound to our Theorem 1.1 in thecase of twisted L -functions for half-integral weight holomorphic cusp forms. In [26], thefirst author and Min Lee were able to obtain a Weyl-type estimate for the Rankin-Selbergconvolution of a holomorphic cusp form and a Maass form in the eigenvalue aspect. Weanticipate that more varied subconvexity results can be achieved via these shifted sums.1.2. A bit more on shifted convolutions. As mentioned above, the main purpose ofthis paper is to present a new method for approaching problems related to shifted sums,subconvexity and moments. In particular, we will introduce a new class of shifted singleand multiple Dirichlet series and show how their full meromorphic continuations can beobtained and how they can be applied to this class of problems.Let (cid:96) , (cid:96) be two positive integers, and fix h ≥ 1. Suppose that f, g are modular forms, ofeven weight k and square-free level N , which have Fourier expansions f ( z ) = (cid:88) m ≥ a ( m ) e πimz = (cid:88) m ≥ A ( m ) m ( k − / e πimz ,g ( z ) = (cid:88) m ≥ b ( m ) e πimz = (cid:88) m ≥ B ( m ) m ( k − / e πimz . (1.3)The constraint that k is even arises from our use of Andre Reznikov’s result in the appendixof this work.The fundamental single Dirichlet series that we will investigate is D ( s ; h, (cid:96) , (cid:96) ) = (cid:88) m ≥ ,m (cid:96) m (cid:96) h a ( m )¯ b ( m )( m (cid:96) ) s + k − . The (cid:96) , (cid:96) parameters are included for use in amplification arguments targeted at upperbounds for individual L -series. See, for example, [5], [7] and [45]. When (cid:96) = (cid:96) = 1 thisspecializes to D ( s ; h ) = (cid:88) m ≥ a ( m + h )¯ b ( m ) m s + k − . (1.4) ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 5 Selberg mentions that the meromorphic continuation of his series can be obtained bycombining “the technique of Rankin” with a certain Poincare series: (cid:88) γ ∈ Γ ∞ \ Γ ( Im ( γz )) s e πihγz . Goldfeld, [17], wrote down the series (1.4), rather than (1.1) and remarked that Selberghad briefly indicated how to obtain its meromorphic continuation. Goldfeld approachesthat continuation in a different way, using instead the Poincare series (cid:88) γ ∈ Γ ∞ \ Γ ( Im γz ) I s − (2 π | h | Im γz ) e πih Re γz . The function I v ( y ) is the modified Bessel function of the first kind, and grows expo-nentially. Goldfeld achieves the meromorphic continuation of (1.4) to Re s > via theanalytic properties of an inner product of his Poincare series with f and g . However theDirichlet series that comes out of this method is more closely related to (1.1) than to (1.4),and a hypergeometric series needs to be introduced to carry over the information. Thepresence of this hypergeometric series seriously impacts efforts to continue past Re s > .A fundamental difficulty with the series (1.1) seems to be the tying together of the h andthe m variables in the denominator. One can see, for example, that if one is interestedin estimating sums of coefficients of a given length x , that this sum becomes seriouslyunbalanced when h greatly exceeds x . It seems to be the case that the best way to “untie”the m and h , so as to avoid unwanted alternative coupling from integral expansions andsmoothing functions, is to reverse the sign of the h in the exponent of the Poincare seriesobtaining, for example: P h ( z ; s ) = (cid:88) γ ∈ Γ ∞ \ Γ ( Im ( γz )) s e − πi h γz . When one does this, the series D ( s ; h ) comes naturally out of a formal computation ofa triple product. Unfortunately, the series P h ( z ; s ) grows exponentially in y , making itdifficult to turn the formal computation into a genuine one. Goldfeld faced a similardifficulty in [17] and dealt with it by truncating the fundamental domain. In the presentcase the difficulties seem to be more fundamental and it appears to be necessary to workwith an approximation to P h ( z ; s ) by two extra parameters. This is done in Sections 2-5.It is shown there that the general series D ( s ; h, (cid:96) , (cid:96) ), which converges for Re s > 1, hasa meromorphic continuation to all of C , with rightmost poles at the points ± it j , wherethe eigenvalues of the Laplacian are + t j . To accomplish this, a spectral expansion for D ( s ; h, (cid:96) , (cid:96) ) is found which is locally normally convergent for Re s < − k . The gapbetween 1 and − k is bridged by an approximation to D ( s ; h ) which is locally normallyconvergent everywhere, and whose limit approaches D ( s ; h ) as the two parameters tendto infinity and zero respectively. The consequent meromorphic continuation of D ( s ; h ) isthe content of Proposition 5.1. JEFF HOFFSTEIN AND THOMAS A. HULSE Most applications of shifted sums call for a sum over the shift h . Because of this it is nat-ural to consider the collective behavior of D ( s ; h, (cid:96) , (cid:96) ) as h varies. This is accomplishedfirst in the meromorphic continuation to { ( s, w ) ∈ C | Re w > } of the shifted doubleDirichlet series: Z ( s, w ) = (cid:88) h ≥ D ( s ; h ) h w +( k − / = (cid:88) h ≥ ,m ≥ m (cid:96) m (cid:96) h a ( m )¯ b ( m )( m ) s + k − h w +( k − / , and more generally of the series Z Q ( s, w ) = (cid:88) h,m ≥ m (cid:96) m (cid:96) hQ a ( m )¯ b ( m )( (cid:96) m ) s + k − ( hQ ) w +( k − / . This series has poles in s at the same locations as D ( s ; h ), but it also has additionalemergent polar lines at w + 2 s + k − ∈ Z ≤ . In Section 8 we prove the following theorem,which is the underpinning of the application. Theorem 1.3. Fix N , (cid:96) , (cid:96) , h ∈ N , with N , (cid:96) , (cid:96) square free and ( N , (cid:96) (cid:96) ) = 1 . Let S Q ( x, y ) = (cid:88) h,m ≥ (cid:96) m (cid:96) m hQ A ( m ) ¯ B ( m ) G ( m /y ) G ( m /x ) , where G ( x ) , G ( x ) are smooth, with compact support in the interval [1 , . Suppose y (cid:29) x (cid:29) , (cid:96) , (cid:96) ∼ L (cid:28) Q . Then S Q ( x, y ) (cid:28) Q θ − + (cid:15) Ly (cid:15) ( y/x ) (cid:15) . The coefficients A ( m ) , B ( m ) are constant on average and the length of this sum is xy/Q .Thus the Theorem gives a square root of the length estimate when y and x are of compa-rable sizes which, if we let Q = 1, matches the trivial cancellation we expect to get if wetreat A ( n ) and B ( n ) like random variables and heuristically factor the double sum as theproduct of two partial sums. The result becomes non-trivial when Q (cid:15) (cid:29) y = x , whichis the case which is relevant to this paper. This result could be considerably improvedif one had further information about upper bounds for Z Q ( s, w ) in the region Re w ≤ f and g are replaced with other automorphicforms. Indeed, by considering the shifted convolution sums of Fourier coefficients of a thetafunction and a half-integral weight Eisenstein series, the second author, Kıral, Kuan, andLi-Mei Lim were able to produce asymptotic counts of binary quadratic forms with squarediscriminants under certain constraints [27].One interesting aspect of the one and two-variable shifted Dirichlet series discussed is thateach has two incarnations: a Dirichlet series that converges locally normally in one range,and a spectral expansion that converges locally normally in a disjoint region. As one ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 7 might expect, all the interesting information seems to be hidden in between in a “criticalstrip” − k < Re s < 1. In particular, the point (1 − k , ) lies inside this critical strip, andthe “correct” bound for S Q ( x, y ) would follow from an accurate estimate for Z Q (1 − k , ).Indeed, by obtaining mean value estimates for Z Q (1 − k , ) as Q varies over a shortinterval, the first author and Min Lee were able to obtain an asymptotic formula for thesecond moment 1 ϕ ( Q ) (cid:88) χ mod Q L (1 / , f, χ ) L (1 / , g, χ ) . By demonstrating that this result is non-zero, they were able to produce a simultaneousnon-vanishing result [25].The spectral expansions yield another curious development. The terms decay polyno-mially in the spectral parameter t j , rather than exponentially, and the ratio of gammafactors governing this decay seems to act in a manner analogous to the usual m − σ indexingof a Dirichlet series (polynomial decay corresponding to the size of σ .)This work is an expanded version of part of a previous work [24] by the first author. Asystematic error in the meromorphic continuation in the continuous part of the spectrumwas observed and corrected by the second author along with strengthening other resultsin sections 3-6, and 8.The authors would like to thank Valentin Blomer, David Hansen, Roman Holowinsky,Mehmet Kıral, Alex Kontorovich, Min Lee, Andre Reznikov, and Matt Young for manyvery helpful and enlightening conversations. We would also like to thank this paper’sreferee, who provided many suggestions that helped us to clarify and streamline many ofour arguments.1.3. An outline of the paper. In Section 2, the Poincare series P h,Y ( z, s ; δ ) is intro-duced and its spectral expansion is computed. A function M ( s, t, δ ) appears in this expan-sion and, in Section 3, the meromorphic continuation of M ( s, t, δ ) is obtained. In Section4, an approximation to D ( s ; h ), namely D ( s ; h, δ ), is introduced, and its meromorphiccontinuation is obtained. The Appendix is an important ingredient in this continuation.In Section 5, the limit as δ → D ( s ; h ). In Section 6, an application to single shiftedsums is given, and in Section 7, the double Dirichlet series Z Q ( s, w ) is introduced andits meromorphic continuation is obtained. In Section 8, an application to double shiftedsums is given, and in Section 9, this double shifted sum is in turn applied to obtain oursubconvexity bound. Remark 1.4. In this paper any use of the term “convergence” shall refer to local normalconvergence. Also, unless otherwise indicated, the implied constant in a (cid:28) , (cid:29) and big- O expression will depend upon a particular (cid:15) appearing in the expression, a constant A ,which is defined in Section 3, and cusp forms f, g , including their level N , but will beindependent of all other variables. Furthermore the notation x ∼ T specifically denotes JEFF HOFFSTEIN AND THOMAS A. HULSE all allowable x found in the interval [ T , T ] and is somewhat different from the usualLandau Notation for asymptotic.2. Spectral expansions Fix N , (cid:96) , (cid:96) , h ∈ N , with N , (cid:96) , (cid:96) square free, ( N , (cid:96) (cid:96) ) = 1 and set N = N (cid:96) (cid:96) / ( (cid:96) , (cid:96) )and Γ = Γ ( N ). The requirement that N be square-free is imposed to simplify theFourier expansions of Eisenstein series that appear in spectral expansions.For z ∈ H and s ∈ C , Re ( s ) > 1, consider the Poincare series: P h ( z ; s ) := (cid:88) γ ∈ Γ ∞ \ Γ ( Im ( γz )) s e − πi h γz . (2.1)This is the usual Poincare series except that the sign in the exponent is negative ratherthan positive. For any fixed z the series converges locally normally where Re ( s ) > y , which complicatesefforts to determine its spectral properties. For this reason, for any fixed Y (cid:29) δ > P h,Y ( z, s ; δ ) by P h,Y ( z, s ; δ ) := (cid:88) γ ∈ Γ ∞ \ Γ ψ Y ( Im γz )( Im ( γz )) s e − πi h Re γz +(2 π h Im γz )(1 − δ ) , (2.2)where ψ Y is the characteristic function of the interval [ Y − , Y ].The series P h,Y ( z, s ; δ ) consists of finitely many terms and is in L (Γ \ H ). It also possessesa Fourier expansion, which can be easily worked out: P h,Y ( z, s ; δ ) = (cid:88) m a m,Y ( s, y ; δ ) e πimx , (2.3)where a m,Y ( s, y ; δ ) = δ m, − h y s e πhy (1 − δ ) ψ Y ( y ) (2.4)+ (cid:88) c (cid:54) =0 S ( m, − h ; cN ) N s | c | s · y − s × (cid:90) ∞−∞ e (2 πihu +2 πh (1 − δ )) / ( N c y ( u +1)) − πimuy ( u + 1) s ψ Y (cid:18) N c y ( u + 1) (cid:19) du. The Petersson inner product of two forms F, G in L (Γ \ H ) is defined by (cid:104) F, G (cid:105) = 1 V (cid:90) (cid:90) Γ \ H F ( z ) G ( z ) dxdyy , (2.5)where V = volume(Γ \ H ) = π N (cid:89) p | N (cid:0) p − (cid:1) . ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 9 Set V to be the function V f,g ( z ; (cid:96) , (cid:96) ) := y k f ( (cid:96) z ) g ( (cid:96) z ) , (2.6)where f and g are as defined in (1.3), which is rapidly decreasing in the cusps of Γ. Wesee that V f,g is Γ-invariant. Indeed, for γ = ( a bNc d ) ∈ Γ, we have that, f ( (cid:96) ( γz )) = f (cid:32)(cid:32) (cid:96) 00 1 (cid:33) (cid:32) a b N (cid:96) (cid:96) ( (cid:96) ,(cid:96) ) c d (cid:33) z (cid:33) (2.7)= f (cid:32)(cid:32) a b(cid:96) N (cid:96) ( (cid:96) ,(cid:96) ) c d (cid:33) (cid:96) z (cid:33) = ( N cz + d ) k f ( (cid:96) z ) , as N (cid:96) ( (cid:96) ,(cid:96) ) ∈ N Z . Similarly, we have g ( (cid:96) ( γz )) = ( N cz + d ) k g ( (cid:96) z ) and so Im ( γz ) k f ( (cid:96) ( γz )) g ( (cid:96) ( γz )) = y k f ( (cid:96) z ) g ( (cid:96) z ) , (2.8)giving the Γ-invariance of V .Our approach will be to compute the inner product (cid:104) P h,Y ( ∗ , s ; δ ) , V (cid:105) in several differentways, and to then let Y → ∞ . To begin, we compute I (cid:96) ,(cid:96) ,Y,δ ( s ; h ) := (cid:104) P h,Y ( ∗ , s ; δ ) , V (cid:105) (2.9)= 1 V (cid:90) (cid:90) Γ \ H P h,Y ( z, s ; δ ) f ( (cid:96) z ) g ( (cid:96) z ) y k dx dyy = 1 V (cid:88) m ≥ (cid:88) m ≥ a ( m )¯ b ( m ) (cid:90) e πix ( m (cid:96) − m (cid:96) − h ) dx × (cid:90) ∞ e − πy ( m (cid:96) + m (cid:96) − h (1 − δ )) y s + k − ψ Y ( y ) dyy = 1 V (4 π ) − ( s + k − (cid:88) m (cid:96) = m (cid:96) + h a ( m )¯ b ( m )( m (cid:96) ) s + k − × (cid:90) Y π(cid:96) m Y − π(cid:96) m e − y (1+ hδ/ (2 (cid:96) m )) y s + k − dyy . Note that for Re s > 1, and all Y > , δ > I (cid:96) ,(cid:96) ,Y,δ ( s ; h ) (cid:28) N Γ( Re s + k − (cid:88) m (cid:96) = m (cid:96) + h | a ( m )¯ b ( m ) | ( m (cid:96) ) Re s + k − , where the implied constant is absolute.Let { u j } j ≥ be an orthonormal basis (with respect to the inner product defined in (2.5))for the discrete part of the spectrum of the Laplace operator on L (Γ \ H ). Suppose the u j have eigenvalue 1 / t j , and Fourier expansion u j ( z ) = (cid:88) n ρ j ( n ) y K it j (2 π | n | y ) e πinx . As P h,Y ( z, s ; δ ) ∈ L (Γ \ H ) we can obtain the spectral expansion of I : I (cid:96) ,(cid:96) ,Y,δ ( s ; h ) = (cid:88) j ≥ (cid:104) V, u j (cid:105) (cid:104) P h,Y ( ∗ , s ; δ ) , u j (cid:105) (2.10)+ 14 π (cid:88) a V (cid:90) ∞−∞ (cid:10) V, E a ( ∗ , + it ) (cid:11) (cid:10) P h,Y ( ∗ , s ; δ ) , E a ( ∗ , + it ) (cid:11) dt. We easily compute (cid:104) P h,Y ( ∗ , s ; δ ) , u j (cid:105) = ρ j ( − h ) V (2 πh ) s − (cid:90) Y πhY − πh y s − e y (1 − δ ) K it j ( y ) dyy (2.11)and (cid:10) P h,Y ( ∗ , s ; δ ) , E a ( ∗ , + it ) (cid:11) = 1 V (2 πh ) s − (cid:32) π + it | h | it ρ a (cid:0) + it, − h (cid:1) Γ (cid:0) + it (cid:1) (cid:33) (cid:90) Y πhY − πh y s − e y (1 − δ ) K it ( y ) dyy . (2.12)Note that u is constant and (cid:104) u , P h,Y (cid:105) = 0. We have taken the Eisenstein series notationfrom [11], that is: E a ( z, s ) denotes the Eisenstein series expanded at the cusp a . Here foreach cusp a = u/w , E a ( z, s ) = δ a , ∞ y s + √ π Γ( s − ) ρ a ( s, y − s Γ( s )+ 2 π s √ y Γ( s ) (cid:88) m (cid:54) =0 | m | s − ρ a ( s, m ) K s − (2 π | m | y ) e πimx , (2.13)and ρ a ( s, m ) = (cid:32) (cid:0) w, Nw (cid:1) wN (cid:33) s (cid:88) ( γ,N/w )=1 γ s (cid:88) δ mod γw ( δ,γw )=1 δγ ≡ u mod( w,N/w ) e − πimδ/ ( γw ) . (2.14)More precisely, since N is square free we have, by [25], for each cusp a = w where w | N that ρ a ( s, n ) = (cid:16) wN (cid:17) s (cid:80) d | n, ( d,N )=1 d − s ζ (2 s ) × (cid:89) p | N (1 − p − s ) − (cid:89) p | w,pk (cid:107) n,k ≥ p − s − p − s +1 (cid:0) p − p k ( − s +1)+1 − p ( k +1)( − s +1) (cid:1) (2.15) ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 11 for n (cid:54) = 0.By Lemma 3.4 in [5] we have that for t ∈ R the ρ a ( s, m ) satisfy (cid:88) a | ρ a ( + it, m ) | (cid:28) (cid:15) ((1 + | t | ) mN ) (cid:15) . (2.16)We will ultimately need to understand the behavior of the integral appearing in (2.11)and (2.12) for fixed small δ as Y → ∞ . With this in mind we define M ( s, t, δ ) = (cid:90) ∞ y s − e y (1 − δ ) K it ( y ) dyy . (2.17)Consider the well-known bounds K γ + ir ( y ) (cid:28) e − y √ y (2.18)for y (cid:29) K γ + ir ( y ) (cid:28) (cid:40) y −| γ | if γ + ir (cid:54) = 0 | log( y ) | if γ + ir = 0 (2.19)for y (cid:28) γ, r ∈ R . From the integral formula K γ + ir ( y ) = 12 (cid:90) ∞ e − y ( u + u − ) u γ + ir duu , (2.20)we see that | K γ + ir ( y ) | ≤ K γ ( y ). Thus we can take the bound (2.18) to be uniform in r for fixed γ , and similarly if γ (cid:54) = 0 we can take the bound (2.19) to be uniform in r as well.Combining this with the recursive formula K ir ( y ) = y ir ( K ir ( y ) + K − ir ( y )) (2.21)we see that the bound (2.19) can be taken to be uniform for γ = 0 and | r | ≥ 1. Given that,it is also clear that the integral expression for M ( s, t, δ ), given in (2.17), converges for any δ > Re s > + | Im t | . (It also converges for δ = 0, when + | Im t | < Re s < M ( s, t, δ ) when Re s > + | Im t | . Proposition 2.1. Let M ( s, t, δ ) be as defined in (2.17) . Fix (cid:15) > , > δ > , Y (cid:29) max( δ − , h ) , A ∈ Z ≥ , t ∈ C and s ∈ C with Re s > + | Im t | + (cid:15) . Then (cid:12)(cid:12)(cid:12)(cid:12) M ( s, t, δ ) − (cid:90) Y πhY − πh y s − e y (1 − δ ) K it ( y ) dyy (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) e − Y πhδ ( Y h ) Re s + A − δ (1 + | t | ) A + ( Y − h ) Re s − −| Im t | (1 + | t | ) A , with the implied constant depending only on A , s , | Im t | and (cid:15) .Proof. We first need to bound K it ( y ) uniformly in t , for y both large and small. For thiswe have Lemma 2.2. For A ∈ Z ≥ and | r | , y (cid:29) , K γ + ir ( y ) (cid:28) A,γ e − y y A √ y (1 + | r | ) A . For A ∈ Z ≥ , | r | (cid:29) , and < y (cid:28) , K γ + ir ( y ) (cid:28) A,γ y −| γ | (1 + | r | ) A . To prove Lemma 2.2 we recall the definition (2.20). The result follows after integratingby parts A times, applying the bound (2.18) when y is large and (2.19) when y is smallto K m + γ + ir ( y ), with | m | ≤ A .Applying Lemma 2.2 we find that (cid:90) ∞ Y πh y s − e y (1 − δ ) K it ( y ) dyy (cid:28) A, | Im t | (cid:90) ∞ Y πh y Re s − e y (1 − δ ) e − y y A − (1 + | t | ) A dyy (cid:28) | t | ) A (cid:90) ∞ Y πh y Re s + A − e − yδ dyy = 1(1 + | t | ) A δ Re s + A − Γ [ Re s + A − , Y πhδ ] , where Γ[ s, x ] := (cid:90) ∞ x e − y y s dyy (cid:28) s e − x x Re s − , as x → ∞ . Thus when Y (cid:29) δ − , (cid:90) ∞ Y πh y s − e y (1 − δ ) K it ( y ) dyy (cid:28) A,s, | Im t | e − Y πhδ ( Y h ) Re s + A − δ (1 + | t | ) A . Similarly, applying the second part of the lemma, (cid:90) Y − πh y s − e y (1 − δ ) K it ( y ) dyy (cid:28) A, | Im t | ( Y − πh ) Re s − −| Im t | (1 + | t | ) A , completing the proof of Proposition 2.1. (cid:3) The analytic continuation of M ( s, t, δ )In this section we will provide a meromorphic continuation of M ( s, t, δ ) to s ∈ C with Re s ≤ + | Im t | . The following proposition locates the poles of M ( s, t, δ ) and gives threedifferent estimates at points, s , where M ( s, t, δ ) is analytic. ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 13 Remark 3.1. In the proposition below, we find that it is easier to work with M ( s, t, δ ) ifwe work with the variable z = it rather than t alone. So we consider instead M ( s, z/i, δ ).Thus the region Re s ≤ + | Im t | becomes Re s ≤ + | Re z | . Furthermore, to simplifyerror growth estimates, we let t = Im z in the statement of the proposition. This differssomewhat from the definition of M ( s, t, δ ), where t can be complex, but since we keep | Im t | small we will find that the error estimates are accurate for both definitions of t andso this notation is justified. Proposition 3.2. Fix (cid:15) and δ such that > (cid:15), δ > and fix A (cid:29) to be a non-integer. The function M ( s, z/i, δ ) has a meromorphic continuation to all s in the half-plane Re s > − A − | Re z | and for z ∈ C where | Re z | < . In this region M ( s, z/i, δ ) is analytic except, in the case when z (cid:54) = 0 , for simple poles at the points s = ± z − (cid:96) for (cid:96) ∈ Z ≥ with residues Res s = ± z − (cid:96) M ( s, z/i, δ ) = ( − (cid:96) √ π (cid:96) ∓ z Γ( ∓ z + (cid:96) )Γ( ± z − (cid:96) ) (cid:96) !Γ( + z )Γ( − z )+ O (cid:96) ((1 + | t | ) ± Re z − e − π | t | δ ) , (3.1) which are meromorphic in z and here t = Im z . There is no error term when (cid:96) = 0 .When z = 0 there are double poles at s = − (cid:96) for (cid:96) ≥ .Considering the singularities as poles in z , we have that Res z = ± ( s + (cid:96) − ) M ( s, z/i, δ ) = ∓ ( − (cid:96) √ π − s Γ(1 − s )Γ(2 s + (cid:96) − (cid:96) !Γ( s + (cid:96) )Γ(1 − s − (cid:96) )+ O (cid:96),σ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) Γ(2 s + (cid:96) )Γ( s + 1) (cid:12)(cid:12)(cid:12)(cid:12) δ (cid:19) (3.2) These residues above have meromorphic continuations to all s in the half-plane specifiedabove. The error terms have poles at s = − − r for r ≥ (cid:96) − as well, but retain thedependence on δ in the residue. There is no error term if (cid:96) = 0 .When s and z are at least (cid:15) away from the poles of M ( s, z/i, δ ) , we have the upper bound M ( s, z/i, δ ) (cid:28) A,(cid:15) (1 + | t | ) Re s − − A (1 + | s | ) A − Re s +3 δ − A e − π | Im s | . (3.3) Restricting the half-plane to a vertical strip by imposing the additional condition Re s ≤ + | Re z | , and further requiring s and z be at least a distance (cid:15) away from poles, we havethat when δ (1 + | t | ) ≤ , M ( s, z/i, δ ) = √ π − s Γ( s − − z )Γ( s − + z )Γ(1 − s )Γ( − z )Γ( + z )+ O A,(cid:15) (cid:16) (1 + | t | ) Re s − (cid:15) (1 + | s | ) − Re s + | Re z | e − π | Im s | δ (cid:15) (cid:17) (3.4) while for δ (1 + | t | ) > M ( s, z/i, δ ) (cid:28) A,(cid:15) (1 + | t | ) σ − (1 + | s | ) A − Re s +3 e − π | Im s | . (3.5) When Re z = 0 and | z | , | Im s | (cid:29) , | s ± z − − m | = (cid:15) > , for (cid:15) small, we have M ( s, z/i, δ ) (cid:28) m (cid:15) − (1 + | s | ) − Re s e − π | Im s | . (3.6) Remark 3.3. We note that M ( s, z/i, δ ) does not have a pole at s = 1, in spite of theappearance of Γ(1 − s ) in (3.4). It is cancelled by the error term in that formula, whichdoes not vanish as δ → Re ( s + z ) > . Proof. As in the statement of the proposition, take | Re z | < , we will assume thisthroughout the proof. From [20] formula 6.621(3) we obtain, when Re s > + | Re z | , M ( s, z/i, δ ) = (cid:90) ∞ y s − e y (1 − δ ) K z ( y ) dyy (3.7)= √ π z Γ( s − + z )Γ( s − − z ) δ s − + z Γ( s ) F ( s − + z, + z ; s ; 1 − δ ) , where F ( α, β ; γ ; x ) is the usual hypergeometric function. By using the transformationformula 9.132(1), also in [20], this expansion becomes M ( s, z/i, δ ) = √ π − s Γ( s − + z )Γ( s − − z )Γ(1 − s )Γ( + z )Γ( − z ) F ( s − + z, s − − z ; s ; δ ) (3.8)+ (cid:0) π (cid:1) δ − s Γ( s − F ( + z, − z ; 2 − s ; δ ) . Since the Taylor expansion for F ( α, β ; γ ; x ) converges for | x | < Re ( α + β − γ ) < Re s < 2, away from the obvious potential poles, we have the followingexpansion, M ( s, z/i, δ ) = π − s Γ( s )Γ(1 − s )Γ( + z )Γ( − z ) ∞ (cid:88) n =0 Γ( s − + z + n )Γ( s − − z + n ) n !Γ( s + n ) (cid:18) δ (cid:19) n (3.9)+ (cid:0) π (cid:1) δ − s Γ( s − − s )Γ( + z )Γ( − z ) ∞ (cid:88) n =0 Γ( + z + n )Γ( − z + n ) n !Γ(2 − s + n ) (cid:18) δ (cid:19) n . From this it is easy to compute residues. First we note that,Res s =1 M ( s, z/i, δ ) = (1 − (cid:16) π (cid:17) ∞ (cid:88) n =0 Γ( + z + n )Γ( − z + n )( n !) Γ( + z )Γ( − z ) (cid:18) δ (cid:19) n = 0 (3.10)and similarly for m ∈ Z > ,Res s = − m M ( s, z/i, δ ) = ( − m π + m Γ( s )Γ( + z )Γ( − z ) ∞ (cid:88) n = m +1 Γ( n − m − + z )Γ( n − m − − z )( n − m − n ! (cid:18) δ (cid:19) n − ( − m π − δ m +1 Γ( + z )Γ( + z ) ∞ (cid:88) n =0 Γ( + z + n )Γ( − z + n ) n !( m + n + 1)! (cid:18) δ (cid:19) n , (3.11) ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 15 where we see the above residue is also zero when we change the index of the first sumabove from n → n + m + 1. So there are no poles for s ∈ Z ≤ . Furthermore we cancompute that Res s = ± z − (cid:96) M ( s, z/i, δ ) = π (cid:96) ∓ z Γ( ± z − (cid:96) )Γ( ∓ z + (cid:96) )Γ( + z )Γ( − z ) (3.12) × (cid:96) (cid:88) n =0 Γ( ± z + n − (cid:96) )( − (cid:96) − n n !( (cid:96) − n )!Γ( ± z + n − (cid:96) ) (cid:18) δ (cid:19) n . We recall Stirling’s formula for the gamma function. For bounded x ∈ R , as y → ∞ , | Γ( x + iy ) | (cid:16) (1 + | y | ) x − e − π | y | , (3.13)where f ( y ) (cid:16) g ( y ) means that f ( y ) (cid:28) g ( y ) and g ( y ) (cid:28) f ( y ). Using this with (3.12), wesee we can take n = 0 as the main term and bound the remainder to get (3.1). We seefrom (3.9) that when z = 0 that are double poles at s = − (cid:96) for (cid:96) ≤ δ → z rather than poles in s , we get that:Res z = ± ( s + (cid:96) − ) M ( s, z/i, δ ) = ∓ π − s Γ( s )Γ(1 − s )Γ( s + (cid:96) )Γ(1 − s − (cid:96) ) (3.14) × (cid:96) (cid:88) n =0 Γ(2 s + (cid:96) + n − − (cid:96) − n n !( (cid:96) − n )!Γ( s + n ) (cid:18) δ (cid:19) n and again taking n = 0 and using Stirling’s formula gives us (3.2).By formula 9.113 in [20], the hypergeometric function in (3.7) can be written as a contourintegral F ( s − + z, + z ; s ; 1 − δ ) = (3.15)Γ( s )Γ( s − + z )Γ( + z ) (cid:32) πi (cid:90) C Γ( s − + u + z )Γ( + u + z )Γ( − u ) (cid:0) δ − (cid:1) u Γ( s + u ) du (cid:33) where C goes from − i ∞ to i ∞ and curves so that the poles of Γ( − u ) lie to the right of C and the poles of Γ( s − + u + z ) , Γ( + u + z ) lie to the left. For Re ( s + z ) > , C can simplybe the vertical line Re u = a , where a is chosen so that max( − Re ( s + z ) , − − Re z )
The contour when Im s = 0 and z = 0For A ≫ A / ∈ Z , fix any R = − Re ( s + z ) ∈ R , such that R / ∈ Z , ( A + R ) / ∈ Z , and A > + | Re s | + | Re z | , and straighten C to the vertical line Re u = A . In the process,poles are passed over at u = ℓ , for 0 ≤ ℓ < A . The residue after passing the pole at u = ℓ is given by √ π z Γ( s − − z ) δ s − + z Γ( + z ) R ( s, z, ℓ ) , (3.10)where R ( s, z, ℓ ) = Γ( s − + z + ℓ )Γ( + z + ℓ ) ! δ − " ℓ ( − ℓ +1 ℓ !Γ( s + ℓ ) . (3.11)Thus M ( s, z/i, δ ) = √ π z Γ( s − − z ) δ s − + z Γ( + z ) N ( s, z, δ ) , Figure 1. The contour when Im s = 0 and z = 0Write Re ( s + z ) = − R , with R ≥ 0. Choose b such that 0 > b > max( R − [ R ] − , − ).The poles of Γ( + u + z ) all lie to the left of Re u = b and the poles of Γ( − u ) all lieto the right of Re u = b . The poles of Γ( s − + u + z ) to the right of Re u = b are at Re u = − Re ( s + z ) − (cid:96) = R − (cid:96) for 0 ≤ (cid:96) ≤ [ R ]. We can make the curve C follow the line Re u = b until Im u = − Im ( s + z ) and then move and weave around the finitely manypoles between Re u = R − [ R ] and Re u = R , continuing to i ∞ along the line Re u = c ,where R < c < [ R ] + 1. (See Figure 1.) We observe that this is still permissible when R ∈ Z , as long as s + z − / ∈ Z ≤ , as the poles of Γ( − u ) and Γ( s − + u + z ) do notoverlap.We continue to require that s + z − / ∈ Z ≤ . For A (cid:29) A / ∈ Z , fix any R = − Re ( s + z )such that A / ∈ Z and ( A + R ) / ∈ Z , and A > + | Re s | + | Re z | + (cid:15) , and straighten C tothe vertical line Re u = A . In the process, poles of the integrand are passed over at u = (cid:96) , ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 17 for 0 ≤ (cid:96) < A . The residue after passing the pole at u = (cid:96) is given by √ π z Γ( s − − z ) δ s − + z Γ( + z ) R ( s, z, (cid:96) ) , (3.16)where R ( s, z, (cid:96) ) = Γ( s − + z + (cid:96) )Γ( + z + (cid:96) ) (cid:0) δ − (cid:1) (cid:96) ( − (cid:96) +1 (cid:96) !Γ( s + (cid:96) ) . (3.17)Thus M ( s, z/i, δ ) = √ π z Γ( s − − z ) δ s − + z Γ( + z ) N ( s, z, δ ) , where N ( s, z, δ ) = − (cid:88) ≤ (cid:96) (cid:15) and + Re ( z + u ) > + A > 0. Inspecting lines (3.16) to (3.18) it is clear that the onlypossible locations of poles of M ( s, z/i, δ ) for + A > Re ( s + z ) > − A are of the form s = ± z − m for m ∈ Z ≥ .Repeat the argument above but now shift to the vertical line Re u = A + R . This timepoles are passed over at u = (cid:96) , for 0 ≤ (cid:96) < A + R . Changing variables, u → u − ( s − + z ),and substituting back into (3.7), we now have, after moving the contour and subtractingthe residue contribution, M ( s, z/i, δ ) = √ π z Γ( s − − z ) δ s − + z Γ( + z ) N ( s, z, δ ) , where N ( s, z, δ ) = − (cid:88) ≤ (cid:96) M ( s, z/i, δ ). Writing s = σ + ir and u = A + iv and recalling that Im z = t , the following upper bounds areeasily confirmed, using (3.13): R ( s, z, (cid:96) ) (cid:28) A,(cid:15) δ − (cid:96) e − π ( | r + t | + | t |−| r | ) (1 + | r + t | ) σ + Re z + (cid:96) − (1 + | t | ) Re z + (cid:96) (1 + | r | ) σ + (cid:96) − , √ π z Γ( s − − z ) δ s − + z Γ( + z ) (cid:28) A,(cid:15) δ R (1 + | r − t | ) σ − Re z − (1 + | t | ) − Re z e − π ( | r − t |−| t | ) , (3.20)and Γ( u )Γ(1 − s + u )Γ( s − + it − u ) (cid:0) δ − (cid:1) u − s + − it Γ( − it + u ) (3.21) (cid:28) A,(cid:15) (1 + | v | ) A − (1 + | v − r | ) A + − σ (1 + | t − v + r | ) σ + Re z − − A δ A + R (1 + | t − v | ) A − Re z e π ( | v | + | v − r | + | r − v + t |−| t − v | ) . Combining the first two, when | t | ≥ | r | one has (cid:88) ≤ (cid:96) r (1 + | v | ) A − σ (1 + | t − v | ) A +1 − σ − Re z e − π ( | v |− | r | ) du (cid:28) A δ − A (1 + | t | ) σ − A − (1 + | r | ) A − σ +1 e − π | r | . (3.24) ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 19 It is easier to show by a similar argument that when | r | ≥ | t | and s and z are at least (cid:15) away from the poles of M ( s, z/i, δ ), thenΓ( s − − z ) δ s − + z Γ( + z ) (cid:90) ( A ) Γ( u )Γ(1 − s + u )Γ( s − + z − u ) (cid:0) δ − (cid:1) u − s + − z du Γ( − z + u ) (cid:28) A,(cid:15) δ − A (1 + | r | ) A − σ +1 e − π | r | . (3.25)Combining these two bounds with (3.22), and noting that when | t | ≤ | r | (1 + | t | ) σ − A − (1 + | r | ) A − σ +3 (cid:29) (1 + | r | ) A − σ +1 , we get the upper bound (3.3).To prove (3.4) and (3.5), we return to the original contour integral in line (3.15). Recallthat for the contour to exist we require − Re ( s + z ) / ∈ Z ≥ . Also recall that | Re z | < .Again we write Re ( s + z ) = − R , but now begin with R > min(0 , − Re z ). Instead ofdeforming the contour by moving it to the right, we instead straighten the contour untilit becomes the line Re u = b , where b is as defined above, and simple poles are passedover at the points u = − s − (cid:96) − z , for 0 ≤ (cid:96) ≤ [ R ]. The residue of the integrand at u = − s − (cid:96) − z is( − (cid:96) Γ(1 − s − (cid:96) )Γ( s − + z + (cid:96) )( δ − − s − z − (cid:96) (cid:96) !Γ( − z − (cid:96) ) . Thus M ( s, z/i, δ ) = C + D, (3.26)where C = [ R ] (cid:88) (cid:96) =0 ( − (cid:96) √ π z Γ( s − − z ) (cid:96) !Γ( + z )Γ( − z − (cid:96) ) (3.27) × Γ(1 − s − (cid:96) )Γ( s − + z + (cid:96) )(2 − δ ) − s − z ( δ − − (cid:96) and D = √ π z Γ( s − − z ) δ s − + z Γ( + z ) (3.28) × πi (cid:90) b + i ∞ b − i ∞ Γ( s − + z + u )Γ( + z + u )Γ( − u ) (cid:0) δ − (cid:1) u Γ( s + u ) du. Fix some small (cid:15) where 0 < (cid:15) < . In the case where Re z > > R > − Re z , thismeans C = 0, but we can shift b to − Re z − + (cid:15) and then we will retrieve the (cid:96) = 0term. Again letting s and z be (cid:15) > R ≥ 0, an application of Stirling’sformula and the binomial theorem to (3.26), (3.27), (3.28) yields M ( s, z/i, δ ) = √ π − s Γ( s − + z )Γ( s − − z )Γ(1 − s )Γ( + z )Γ( − z )+ O A,(cid:15) (cid:18) (1 + | t | ) σ − (1 + | s | ) − σ + | Re z | e − π | Im s | max (cid:96) =1 , [ R (cid:99) (cid:0) δ (cid:96) (1 + | t | ) (cid:96) (cid:1)(cid:19) + O A,(cid:15) (cid:16) (1 + | t | ) − − Re z − b (1 + | s | ) − σ + | Re z | e − π | Im s | δ − σ − z − b (cid:17) , (3.29)where the main term comes from the (cid:96) = 0 part of C . By shifting the line of integration of D , we can also get roughly the expression when Re z > ≥ R ≥ − Re z , replacing b with − Re z − + (cid:15) and the first error term given above is eliminated.Now consider the two cases: δ (1 + | t | ) ≤ δ (1 + | t | ) > 1. Choose any 1 > (cid:15) > 0. Inthe first case max (cid:96) =1 , [ R (cid:99) (cid:0) δ (cid:96) (1 + | t | ) (cid:96) (cid:1) = δ (1 + | t | ) ≤ δ (cid:15) (1 + | t | ) (cid:15) . The first error term of (3.29) is then (cid:28) (1 + | t | ) σ − (cid:15) (1 + | s | ) − σ + | Re z | e − π | Im s | δ (cid:15) and the second is (cid:28) (1 + | t | ) − (1 + | s | ) − σ + | Re z | e − π | Im s | δ − σ + (cid:15) − (cid:15) which is (cid:28) (1 + | t | ) σ − (cid:15) (1 + | s | ) − σ + | Re z | a e − π | Im s | δ (cid:15) . Thus both pieces of the error term are O (cid:16) (1 + | t | ) σ − (cid:15) (1 + | s | ) − σ + | Re z | e − π | Im s | δ (cid:15) (cid:17) , giving us (3.4).In the second case, δ (1 + | t | ) > 1, we return to the first estimate, (3.3). Substituting δ − A < (1 + | t | ) A we obtain M ( s, z/i, δ ) (cid:28) (1 + | t | ) σ − (1 + | s | ) A − σ +3 e − π | Im s | , giving us (3.5).Let r = Im s . Let Re z = 0, and | t | , | r | (cid:29) | s + z − − m | = (cid:15) forsmall (cid:15) > 0, which forces t ∼ − r . Then using (3.26), Stirling’s Approximation andΓ( s − + z + m ) (cid:28) m (cid:15) − , we have that M ( s, z/i, δ ) (cid:28) m (cid:15) − e − π | r | . (3.30) ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 21 Similarly suppose that | s − z − − m | = (cid:15) > 0, which forces t ∼ r . Again using (3.26),Stirling’s Approximation, and Γ( s − − z ) (cid:28) m (cid:15) − , we have that M ( s, z/i, δ ) (cid:28) m (cid:15) − (1 + | r | ) − σ e − π | r | (3.31)which together with (3.30) gives (3.6), completing the proof of Proposition 3.2. (cid:3) The limit as Y → ∞ We are now in a position to analyze the uniformity of convergence of the spectral decom-position of I (cid:96) ,(cid:96) ,Y,δ ( s ; h ) given in (2.10). We assume at first that Re s > . Recall thatby (2.10) we may write I (cid:96) ,(cid:96) ,Y,δ ( s ; h ) = I cusp (cid:96) ,(cid:96) ,Y,δ ( s ; h ) + I cts (cid:96) ,(cid:96) ,Y,δ ( s ; h ) , where I cusp (cid:96) ,(cid:96) ,Y,δ ( s ; h ) = (cid:88) j ρ j ( − h ) V (2 πh ) s − (cid:90) Y πhY − πh y s − e y (1 − δ ) K it j ( y ) dyy (cid:104) V, u j (cid:105) and I cts (cid:96) ,(cid:96) ,Y,δ ( s ; h ) = 14 π V (2 πh ) s − (cid:88) a (cid:90) ∞−∞ (cid:32) V π + it | h | it ρ a (cid:0) + it, − h (cid:1) Γ (cid:0) + it (cid:1) (cid:33) × (cid:90) Y πhY − πh y s − e y (1 − δ ) K it ( y ) dyy (cid:10) V, E a ( ∗ , + it ) (cid:11) dt. Of key importance is the average size of ρ j ( − h ) and of (cid:104) V, u j (cid:105) . The second is addressedby the following proposition, a generalization of [4], which is proved in the Appendix byAndre Reznikov (Theorem B): Proposition 4.1. With the notation as given above (cid:88) a (cid:90) T − T V (cid:12)(cid:12)(cid:10) V, E a ( ∗ , + it ) (cid:11)(cid:12)(cid:12) e π | t | dt + (cid:88) | t j |∼ T | (cid:104) V, u j (cid:105) | e π | t j | (cid:28) ( (cid:96) (cid:96) ) − k T k log( T ) . Remark 4.2. An explicit analysis by Hansen, [22], of some formulas of Ichino, [28],reveals that Proposition 4.1 is essentially stating that L ( , f ⊗ ¯ g ⊗ ¯ u j ) L (1 , Ad u j ) (4.1)is close to constant on average. An application of the Kuznetsov trace formula (see, for example, [29], Theorem 16.8, line(16.56)) shows that (cid:88) | t j |∼ T | ρ j ( − h ) | e − π | t j | (cid:28) (cid:96) (cid:96) T + T / σ ( h ) σ ( (cid:96) (cid:96) )( h, (cid:96) (cid:96) ) h log (3 h ) . (4.2)Alternatively, for ( h, (cid:96) (cid:96) N ) = 1, one can factor ρ j ( − h ) = ρ j ( − λ j ( h ), where λ j ( h ) isthe Hecke eigenvalue at h of the new form associated to u j , of level dividing N . Note thatif u j is an old form it is possible that ρ j ( − 1) = 0. One then has (cid:88) | t j |∼ T | ρ j ( − h ) | e − π | t j | (cid:28) h θ (cid:88) | t j |∼ T | ρ j ( − | e − π | t j | (cid:28) h θ (cid:96) (cid:96) T , (4.3)where θ represents the best progress toward the (non-archimedean) Ramanujan conjec-ture.Proposition 4.1, Cauchy-Schwarz, and (4.2) give us (cid:88) | t j |∼ T ρ j ( − h ) (cid:104) V, u j (cid:105) (cid:28) (cid:88) | t j |∼ T | ρ j ( − h ) | e − π | t j | (cid:88) | t j |∼ T | (cid:104) V, u j (cid:105) | e π | t j | (4.4) (cid:28) (cid:16) (cid:96) (cid:96) T + T / σ ( (cid:96) (cid:96) )( h, (cid:96) (cid:96) ) h log (3 h ) (cid:17) (cid:0) ( (cid:96) (cid:96) ) − k T k log( T ) (cid:1) (cid:28) ( (cid:96) (cid:96) ) − k T k log( T ) + ( (cid:96) (cid:96) ) − k T / k (cid:16) σ ( (cid:96) (cid:96) )( h, (cid:96) (cid:96) ) h log (3 h ) log( T ) (cid:17) . Note that for (cid:96) , (cid:96) prime, (cid:96) , (cid:96) (cid:29) h / log(3 h ), with ( h, (cid:96) (cid:96) ) = 1, the first of the twoterms on the right hand side of (4.4) dominates.We summarize the above discussion in the following proposition. Proposition 4.3. For ( h, N ) = 1 , (cid:88) | t j |∼ T ρ j ( − h ) (cid:104) V, u j (cid:105) (cid:28) h θ ( (cid:96) (cid:96) ) − k T k log( T ) . In addition, (cid:88) | t j |∼ T ρ j ( − h ) (cid:104) V, u j (cid:105)(cid:28) ( (cid:96) (cid:96) ) − k T k log( T ) + ( (cid:96) (cid:96) ) − k T / k (cid:16) σ ( (cid:96) (cid:96) )( h, (cid:96) (cid:96) ) h log (3 h ) log( T ) (cid:17) . Similarly, Proposition 4.1, Cauchy-Schwarz, and (2.16) imply that (cid:90) T − T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) a V π − it | h | − it ρ a (cid:0) − it, − h (cid:1) (cid:10) V, E a ( z, + it ) (cid:11) Γ( − it ) ζ (1 + 2 it ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt (cid:28) ( (cid:96) (cid:96) ) − k + (cid:15) T k + + (cid:15) . (4.5) ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 23 Recalling (2.10), (2.11) and (2.12), we define I (cid:96) ,(cid:96) ,δ ( s ; h ) for Re s > by I (cid:96) ,(cid:96) ,δ ( s ; h ) = I cusp (cid:96) ,(cid:96) ,δ ( s ; h ) + I cts (cid:96) ,(cid:96) ,δ ( s ; h ) , (4.6)where I cusp (cid:96) ,(cid:96) ,δ ( s ; h ) = (cid:88) j ρ j ( − h ) V (2 πh ) s − M ( s, t j , δ ) (cid:104) V, u j (cid:105) (4.7)and I cts (cid:96) ,(cid:96) ,δ ( s ; h ) = 14 π (cid:88) a (cid:90) ∞−∞ V (2 πh ) s − (cid:32) V π − it | h | − it ρ a (cid:0) − it, − h (cid:1) Γ (cid:0) − it (cid:1) (cid:33) × M ( s, t, δ ) (cid:10) V, E a ( ∗ , + it ) (cid:11) dt. (4.8)By the first upper bound of Proposition 3.2, (3.3), for any fixed s and δ , with s at least adistance (cid:15) from the poles, M ( s, t, δ ) decays faster than any power of | t | as | t | → ∞ . Usingthis, another application of Proposition 4.1 and Cauchy-Schwarz, we see that the integralover t in (4.8) converges for any fixed s such that Re s (cid:54) = − n for n ≥ 0. However, theintegral representation of I cts (cid:96) ,(cid:96) ,δ ( s ; h ) only holds for when Re s > , and so we investigateits meromorphic continuation to all Re s > − A . Changing variables from t → z/i , werewrite I cts (cid:96) ,(cid:96) ,δ ( s ; h ) as I cts (cid:96) ,(cid:96) ,δ ( s ; h ) (4.9)= (cid:88) a πi (cid:90) (0) V (2 πh ) s − (cid:32) V π − z | h | − z ρ a (cid:0) − z, − h (cid:1) Γ (cid:0) − z (cid:1) ζ ∗ (1 + 2 z ) (cid:33) M ( s, z/i, δ ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) dz. where ζ ∗ ( z ) = π − z Γ( z ) ζ ( z ) is the completed Riemann zeta function and E ∗ a ( s, z ) = ζ ∗ (2 z ) E a ( s, z ). Consider s in the region of C to the right of Re s = and left of + C ,where C is a curve of the form C ( x ) = c log(2+ | x | ) + ix for some constant c > ζ ∗ (1 + 2 s ) has no zeros on, or to the right of, − C . We can then shift the line of integrationin (4.9) to the left, moving over a pole at z = − s to get I cts (cid:96) ,(cid:96) ,δ ( s ; h ) = (cid:88) a (cid:34) πi (cid:90) − C V (2 πh ) s − (cid:32) V π − z | h | − z ρ a (cid:0) − z, − h (cid:1) Γ (cid:0) − z (cid:1) ζ ∗ (1 + 2 z ) (cid:33) M ( s, z/i, δ ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) dz + 1 V (2 πh ) s − (cid:32) V π s | h | − + s ρ a ( s, − h )Γ ( s ) ζ ∗ (2 − s ) (cid:33) √ π − s Γ(2 s − s ) (cid:104) V, E ∗ a ( ∗ , − s ) (cid:105) (cid:35) . (4.10)The residue is obtained from (3.2) in Proposition 3.2. We see that (4.10) defines a mero-morphic function for s between − C and + C . So shifting s left such that it is between − C and (0), we can shift the line of integration for z back to 0 past the pole at z = s − . This gives us I cts (cid:96) ,(cid:96) ,δ ( s ; h ) =1 V (2 πh ) s − (cid:88) a (cid:34) πi (cid:90) (0) (cid:32) V π − z | h | − z ρ a (cid:0) − z, − h (cid:1) Γ (cid:0) − z (cid:1) ζ ∗ (1 + 2 z ) (cid:33) M ( s, z/i, δ ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) dz + (cid:32) V π − s | h | − s + ρ a (1 − s, − h )Γ(1 − s ) ζ ∗ (2 s ) (cid:104) V, E ∗ a ( ∗ , s ) (cid:105) + V π s | h | − + s ρ a ( s, − h )Γ ( s ) ζ ∗ (2 − s ) (cid:104) V, E ∗ a ( ∗ , − s ) (cid:105) (cid:33) × √ π − s Γ(2 s − s ) (cid:35) for − < Re s < . This argument can be reproduced as we pass over every line at s = − n , and so proceeding recursively, we get that for − − n < Re s ≤ − n , I cts (cid:96) ,(cid:96) ,δ ( s ; h ) (4.11)= 1 V (2 πh ) s − (cid:88) a πi (cid:90) − C σ (cid:32) V π − z | h | − z ρ a (cid:0) − z, − h (cid:1) Γ (cid:0) − z (cid:1) ζ ∗ (1 + 2 z ) (cid:33) M ( s, z/i, δ ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) dz + (cid:98) − σ (cid:99) (cid:88) (cid:96) =0 (cid:32) V π − s − (cid:96) | h | − s − (cid:96) + ρ a (1 − s − (cid:96), − h )Γ(1 − s − (cid:96) ) ζ ∗ (2 s + 2 (cid:96) ) (cid:104) V, E ∗ a ( ∗ , s + (cid:96) ) (cid:105) + (1 − δδδ σ,(cid:96) ) V π s + (cid:96) | h | − + s + (cid:96) ρ a ( s + (cid:96), − h )Γ ( s + (cid:96) ) ζ ∗ (2 − s − (cid:96) ) (cid:104) V, E ∗ a ( ∗ , − s − (cid:96) ) (cid:105) (cid:33) × ( − (cid:96) √ π − s Γ(1 − s )Γ(2 s + (cid:96) − (cid:96) !Γ( s + (cid:96) )Γ(1 − s − (cid:96) ) (1 + O σ ( δ )) (cid:35) where C σ = (cid:40) (0) when − σ / ∈ Z ≥ C when − σ ∈ Z ≥ (4.12)and δδδ σ,(cid:96) = (cid:40) (cid:96) (cid:54) = − σ (cid:96) = − σ. (4.13)The dependence on σ in the error term arises from our estimates of the residues of M ( s, z/i, δ ) in (3.2) and the dependence on the location of the poles. From what weknow about (3.2) in Proposition 3.2, the error term above also has a meromorphic contin-uation to all s in the strip we are working in, and we see it has, at most, the same polesas the residual term, though the residues of these poles will vanish as δ → ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 25 Proposition 4.3 combined with (3.3) shows that the right-hand side of (4.7) converges.By the argument given above, I cts (cid:96) ,(cid:96) ,δ ( s ; h ) also has a meromorphic continuation to all Re s > − A . The same argument, together with Proposition 2.1, shows that choosing A sufficiently large will ensure that the expression for I (cid:96) ,(cid:96) ,Y,δ ( s ; h ) in (2.10) converge forany fixed Y , as long as Re s > + θ , and that the difference, |I (cid:96) ,(cid:96) ,δ ( s ; h ) − I (cid:96) ,(cid:96) ,Y,δ ( s ; h ) | ,converges as well, allowing the interchange of the limit as Y → ∞ with the summation.As the limit of each term in the summation is 0 as Y → ∞ this shows that for fixed δ > s with Re s > + θ , I (cid:96) ,(cid:96) ,δ ( s ; h ) is an analytic function of s and I (cid:96) ,(cid:96) ,δ ( s ; h ) = lim Y →∞ I (cid:96) ,(cid:96) ,Y,δ ( s ; h ) . Recall that in (2.9) we had an expression for I (cid:96) ,(cid:96) ,Y,δ ( s ; h ) as a Dirichlet series. For Re s > Y → ∞ , namely theanalytic functionlim Y →∞ I (cid:96) ,(cid:96) ,Y,δ ( s ; h ) = 1 V (4 π ) − ( s + k − (cid:88) m (cid:96) = m (cid:96) + h a ( m )¯ b ( m )( m (cid:96) ) s + k − (cid:90) ∞ e − y (cid:16) δh (cid:96) m (cid:17) y s + k − dyy = Γ( s + k − π ) s + k − V D ( s ; h, δ ) , where D ( s ; h, δ ) := (cid:88) m (cid:96) = m (cid:96) + h a ( m )¯ b ( m )( m (cid:96) ) s + k − (cid:18) δh (cid:96) m (cid:19) − ( s + k − (4.14)converges for Re s > I (cid:96) ,(cid:96) ,δ ( s ; h ) defined by (4.6) is an infinite sumthat converges when Re s > and is related to D ( s ; h, δ ) by D ( s ; h, δ ) := (4 π ) s + k − V Γ( s + k − I (cid:96) ,(cid:96) ,δ ( s ; h ) , (4.15)when Re s > D ( s ; h, δ ) is related to that of I (cid:96) ,(cid:96) ,δ ( s ; h ) via (4.15).Using (4.6),(4.7), (4.8) we obtain a spectral expansion when Re s = σ > − A . For Re ( s ) > , we have D ( s ; h, δ ) = D cusp ( s ; h, δ ) + D cts ( s ; h, δ ) , (4.16)where D cusp ( s ; h, δ ) := (4 π ) k s − √ π Γ( s + k − h s − (cid:88) j ρ j ( − h ) M ( s, t j , δ ) (cid:104) V, u j (cid:105) (4.17)and D cts ( s ; h, δ ) := D int ( s ; h, δ ) + Ω( s ; h, δ ) , (4.18)where, as with (4.11), we have that D int ( s ; h, δ ) is the integral component, D int ( s ; h, δ ) := (4 π ) k s − √ π Γ( s + k − h s − × (cid:88) a πi (cid:90) − C σ V π − z ρ a (cid:0) − z, − h (cid:1) | h | − z Γ (cid:0) − z (cid:1) ζ ∗ (1 + 2 z ) M ( s, z/i, δ ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) dz (4.19)and Ω( s ; h, δ ) arises from the residues of poles,Ω( s ; h, δ ) := (4 π ) k s + k − × (cid:88) a (cid:98) − σ (cid:99) (cid:88) (cid:96) =0 (cid:18) π − s − (cid:96) h − s − (cid:96) V ρ a (1 − s − (cid:96), − h )Γ(1 − s − (cid:96) ) ζ ∗ (2 s + 2 (cid:96) ) (cid:104) V, E ∗ a ( ∗ , s + (cid:96) ) (cid:105) + (1 − δδδ σ,(cid:96) ) π s + (cid:96) h (cid:96) V ρ a ( s + (cid:96), − h )Γ ( s + (cid:96) ) ζ ∗ (2 − s − (cid:96) ) (cid:104) V, E ∗ a ( ∗ , − s − (cid:96) ) (cid:105) (cid:19) × ( − (cid:96) Γ(1 − s )Γ(2 s + (cid:96) − (cid:96) !Γ( s + (cid:96) )Γ(1 − s − (cid:96) ) (1 + O σ ( δ )) (cid:21) , (4.20)and is zero if σ > . Here C σ and δδδ σ,(cid:96) are as in (4.12) and (4.13).By (3.3), (4.4), (4.5), (4.6), for any sufficiently large fixed A (cid:29) 1, the series expressionsfor I cusp (cid:96) ,(cid:96) ,δ ( s ; h ), I cts (cid:96) ,(cid:96) ,δ ( s ; h ), and D ( s ; h, δ ) given in (4.7),(4.8), and (4.16) converge for s with Re s > − A , but the upper bound can have a factor of δ − A in it. As we intend tolet δ → Re s in which there is absolute convergencewith an upper bound that is independent of δ .Take Im t j = 0. Combining (3.4) and (3.5) we have, for − A < Re s = σ ≤ , regardlessof the relation between δ and (1 + | t | ) − , M ( s, t j , δ ) (cid:28) A,(cid:15) (1 + | t j | ) σ − (cid:15) (1 + | s | ) A − σ +3 e − π | Im s | , (4.21)when s is at least (cid:15) away from the poles at s = ± it j − r for r ≥ 0. If Im t j (cid:54) = 0 forsome t j then by the same reasoning a comparable bound holds in the s aspect and we areunconcerned with the growth in the t j aspect from a finite number of terms as they will notaffect convergence of the sum over t j . Combining this with Proposition 4.3 and noting that ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 27 t j ∈ R for sufficiently large j , we have, for any T, | s | (cid:29) 1, with min m ∈ Z | s − ± t j − m | > (cid:15) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) | t j |∼ T ρ j ( − h ) M ( s, t j , δ ) (cid:104) V, u j (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) A,(cid:15) (1 + | s | ) A − σ +3 e − π | Im s | (cid:104) ( (cid:96) (cid:96) ) − k T k +2 σ − (cid:15) log( T ) (4.22)+( (cid:96) (cid:96) ) − k T − / k +2 σ +2 (cid:15) (cid:16) σ ( (cid:96) (cid:96) )( h, (cid:96) (cid:96) ) h log (3 h ) log( T ) (cid:17) (cid:21) . When T is small, a similar bound holds in all aspects except for T . Indeed (4.21) isgenerally unchanged in the s -aspect, with some polynomial growth dependent on A andthe same exponential decay, Furthermore, Proposition 4.3 is uniform and thus unchangedin the (cid:96) , (cid:96) and h aspects even when T is small.Applying (4.5) in place of the proposition, we obtain (cid:90) T − T (cid:88) a V ρ a (cid:0) − it, − h (cid:1) ζ (1 + 2 it ) M ( s, t, δ ) (cid:10) V, E a ( ∗ , + it ) (cid:11) dt (cid:28) ( (cid:96) (cid:96) ) − k + (cid:15) T σ − / k +2 (cid:15) e − π | Im s | (1 + | s | ) A − σ +3 (4.23)when once more s is at least (cid:15) away from any poles of M ( s, t, δ ).Breaking up the interval from 1 to T dyadically and letting T → ∞ it is clear from (4.22)that the cuspidal part of the spectral expansion of D ( s ; h, δ ) converges when − A < Re s < − k − (cid:15) . Referring to (4.23), we see that the continuous part also converges inthis region and satisfies a δ -independent upper bound. Thus (4.22), together with (4.23),(4.19) and Stirling’s formula provide the upper bound for D ( s ; h, δ ) of Proposition 4.4stated below in (4.24).The details of the above discussion are summarized, and some additional information isadded, in the following Proposition 4.4. The function D ( s ; h, δ ) , defined in (4.14) , has a meromorphic contin-uation to the half plane Re s > − A . For Re ( s ) > , we have the decomposition D ( s ; h, δ ) = D cusp ( s ; h, δ ) + D cts ( s ; h, δ ) , where D cusp ( s ; h, δ ) and D cts ( s ; h, δ ) are as in (4.17) and (4.18) respectively, are locallynormally convergent as sums and integrals away from the poles of M ( s, t, δ ) and havemeromorphic continuations to Re ( s ) > − A .If t j (cid:54) = 0 for all j , D cusp ( s ; h, δ ) has simple poles at s = ± it j − r for each t j and each ≤ r < A ∓ Im t j . Furthermore D cts ( s ; h, δ ) has poles when s = − r for A > r ≥ ,and at s = (cid:37) − r for A + Re ( (cid:37) ) − > r ≥ where (cid:37) is a non-trivial zero of ζ ( s ) . Indeed, let R ( s ; h, δ ) = Res s = s D ( s ; h, δ ) then we have R ( ± it j − r ; h, δ )= ( − r (4 π ) k h r ∓ it j ρ j ( − h ) (cid:104) V, u j (cid:105) Γ( ∓ it j + r )Γ( ± it j − r )2 r !Γ( + it j )Γ( − it j )Γ( k − ± it j − r )+ O (cid:16) (1 + | t j | ) r h r ρ j ( − h ) (cid:104) V, u j (cid:105) δ (cid:17) . If there does exist some t j = 0 then we also have double poles at s = − r for r ≥ .In the region − A + (cid:15) < Re s = σ < − k , with s at least a distance (cid:15) > from poles ± it j − r , and | Im s | (cid:29) , D ( s ; h, δ ) satisfies the upper bound D ( s ; h, δ ) − Ω( s ; h, δ ) (cid:28) (1 + | s | ) A (cid:48) ( (cid:96) (cid:96) ) (1 − k ) / h − σ × max (cid:32) , ( h, (cid:96) (cid:96) ) / σ ( (cid:96) (cid:96) ) h / log(3 h )( (cid:96) (cid:96) ) (cid:33) . (4.24) Here A (cid:48) depends only on A . Also, when ( h, N ) = 1 , the term max ( ∗ ) can be replaced by h θ , where θ represents the best progress toward the Ramanujan conjecture.For ≥ σ ≥ − A the upper bound in the above expression is multiplied by δ − A .For σ > (cid:15), D ( s ; h, δ ) satisfies the bound D ( s ; h, δ ) (cid:28) (cid:16) h(cid:96) (cid:17) ( k − / ( (cid:96) (cid:96) ) ( k − / (cid:96) (cid:15) (1 + | s | ) A (cid:48) . (4.25) Proof. Proposition 3.2, (3.3), states that for sufficiently large A , and fixed δ > s satisfying Re s > − A at least a distance (cid:15) from the poles, the meromorphic continuationof M ( s, t, δ ) decays faster than any power of | t | . Thus for any fixed δ > s with Re s > − A , the series expression for I (cid:96) ,(cid:96) ,δ ( s ; h ) given in (4.6) converges, giving ameromorphic continuation of I (cid:96) ,(cid:96) ,δ ( s ; h ) back to Re s > − A , with possible poles at thepoints specified. The corresponding meromorphic continuation of D ( s ; h, δ ) follows from(4.15). The residues at these points of I (cid:96) ,(cid:96) ,δ ( s ; h ), and consequently of D ( s ; h, δ ), followfrom the corresponding residue evaluations of M ( s, t, δ ) given in Proposition 3.2, as wellas by examining properties of (4.19).We observe that the piece corresponding to the continuous part of the spectrum, I cts (cid:96) ,(cid:96) ,δ ( s ; h ),has poles due to the terms Γ(2 s + (cid:96) − ζ ∗ (2 s + 2 (cid:96) )Γ( s + (cid:96) )and it is a straightforward exercise to compute these residues. The bounds on the discretecontinuous parts of D ( s ; h, δ ) for − A < σ < − k have been explained above. Thebound for σ > (cid:15) is immediate. (cid:3) ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 29 The limit as δ → Proposition 5.1. For Re s > , let D ( s ; h ) = lim δ → D ( s ; h, δ ) = (cid:88) m (cid:96) = m (cid:96) + h a ( m )¯ b ( m )( m (cid:96) ) s + k − . (5.1) The function D ( s ; h ) converges for Re s > and has a meromorphic continuation to all s with Re s > − A . If t j (cid:54) = 0 for all j , this function has simple poles at s = ± it j − r for each t j and each ≤ r < A ∓ Im t j . Furthermore it has possible poles, due to Ω( s ; h ) as in (5.15) , when s = − r for A > r ≥ , and at s = (cid:37) − r for A + Re ( (cid:37) ) − > r ≥ where (cid:37) is a non-trivial zero of ζ ( s ) . In particular R ( + it j − r ; h ) = Res s = + it j − r D ( s ; h ) = c r,j h r − it j ρ j ( − h ) , (5.2) where now the index j identifies our choice of + it j or − it j and c r,j = ( − r (4 π ) k (cid:104) V, u j (cid:105) Γ( − it j + r )Γ(2 it j − r )2 r !Γ( + it j )Γ( − it j )Γ( k − + it j − r ) . (5.3) For T (cid:29) , if (cid:96) , (cid:96) are prime, and (cid:96) , (cid:96) (cid:29) h / log(3 h ) , with ( h, (cid:96) (cid:96) ) = 1 , then the c r,j satisfy the average upper bounds (cid:88) | t j |∼ T | c r,j | e π | t j | (cid:28) log( T )( (cid:96) (cid:96) ) − k T r +1 . (5.4) If there exists t j = 0 , then all of the above is true but D ( s ; h ) instead has double poles at s = − r for r ≥ , r ∈ Z .For s in the vertical strip − A < σ < − k , D ( s ; h ) can be expressed by the convergentspectral expression given in (5.14) with upper bounds given by (5.16) . For s in the verticalstrip c < σ < , at least (cid:15) away from the poles noted above, for any c with − A + (cid:15) 2, as follows.Set T (cid:29) 1. Then when | Im s | < T , D cusp ( s ; h, δ ) + (cid:88) | t j | 0, which is permitted by Weyl’s Law. Thus by theupper bounds given in (3.6) and Proposition 4.4, we have that, for T large I (cid:28) c,N,h,s T A (cid:48) e − T ( δ − A + cT ) (5.6)From this it is obvious that I ( T ) → T → ∞ . Similarly lim T →∞ I ( T ) = 0. Thisgives us the integral expression for D cusp ( s ; h, δ ), D cusp ( s ; h, δ ) = 12 πi (cid:90) (2) D cusp ( u ; h, δ ) e ( u − s ) duu − s − πi (cid:90) ( c ) D cusp ( u ; h, δ ) e ( u − s ) duu − s (5.7) − (cid:88) t j , ≤ r< ∓ Im ( t j ) − c R ( + it j − r ; h, δ ) e ( + it j − r − s ) + it j − r − s . Now we observe that for < σ < πi (cid:90) (2) D cts ( u ; h, δ ) e ( u − s ) duu − s (5.8)= (cid:88) a (cid:18) πi (cid:19) (cid:90) (cid:90) (2)(0) V π k − z ρ a ( − z, − h ) h − u − z M ( u, z/i, δ ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) − k − u Γ( u + k − − z ) ζ ∗ (1 + 2 z )( u − s ) e ( u − s ) dzdu where D cts ( u ; h, δ ) is as in (4.18). Since the double integral is convergent in u and z , wecan change the order of integration and shift the line of integration of u left from Re u = 2 ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 31 to Re u = c . This gives us that12 πi (cid:90) (2) D cts ( u ; h, δ ) e ( u − s ) duu − s = D cts ( s ; h, δ ) (5.9)+ (cid:88) a (cid:18) πi (cid:19) (cid:90) (cid:90) ( c )(0) V π k − z ρ a ( − z, − h ) h − u − z M ( u, z/i, δ ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) − k − u Γ( u + k − − z ) ζ ∗ (1 + 2 z )( u − s ) e ( u − s ) dzdu + (cid:98) − c (cid:99) (cid:88) r =0 (cid:88) a πi (cid:90) (0) V π − z ρ a ( − z, − h ) h r (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) − r r !(4 π ) − k Γ( + z )Γ( − z ) ζ ∗ (1 + 2 z ) × (cid:34) h − z Γ( − z + r )Γ(2 z − r ) e ( + z − r − s ) Γ( k − + z − r )( + z − r − s ) + Γ( + z + r )Γ( − z − r ) e ( − z − r − s ) Γ( k − − z − r )( − z − r − s ) (cid:35) dz + O s,c ( δ ) . We note that we can disregard the issue of double poles at z = 0. Indeed, when we movethe line of integration for u near Re u = − r , we can shift the line of integration for z slightly to the left of Re z = 0 into the zero-free region of ζ ∗ (1 + 2 z ) and then move theline for u past simple poles at u = ± z − r . Once we have moved past these poles we canshift the line of integration for z back to Re z = 0 without consequence. Furthermore,we observe the integrands of residual terms in the above expression are well-defined when z = 0 and that these residual terms are also well-defined.Rewriting the above equality we get D cts ( s ; h, δ ) = 12 πi (cid:90) (2) D cts ( u ; h, δ ) e ( u − s ) duu − s (5.10) − πi (cid:90) ( c ) ( D cts ( u ; h, δ ) − Ω( u ; h, δ )) e ( u − s ) duu − s − (cid:98) − c (cid:99) (cid:88) r =0 (cid:88) a πi (cid:90) (0) V π − z ρ a ( − z, − h ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) − r r !(4 π ) − k Γ( + z )Γ( − z ) ζ ∗ (1 + 2 z ) × (cid:34) h r − z Γ( − z + r )Γ(2 z − r ) e ( + z − r − s ) Γ( k − + z − r )( + z − r − s ) + h r Γ( + z + r )Γ( − z − r ) e ( − z − r − s ) Γ( k − − z − r )( − z − r − s ) (cid:35) dz − O s,c ( δ ) , where Ω( s ; h, δ ) is as in (4.20), when < σ < 2. We see that the integrands of the firsttwo integrals have no poles in s when c < σ < D cts ( s ; h, δ ). Indeed, wesee that we only need be concerned with the case when s − + r is near negative integralvalues. By an argument analogous to the one that gave us the continuation of D cts ( s ; h, δ ) in Proposition 4.4, we are able to shift the lines of integration for z left and then backagain as the line of integration for s moves past to get D cts ( s ; h, δ ) = 12 πi (cid:90) (2) D cts ( u ; h, δ ) e ( u − s ) duu − s (5.11) − πi (cid:90) ( c ) ( D cts ( u ; h, δ ) − Ω( u ; h, δ )) e ( u − s ) duu − s − (cid:98) − c (cid:99) (cid:88) r =0 (cid:88) a πi (cid:90) C σ V π − z ρ a ( − z, − h ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) − r r !(4 π ) − k Γ( + z )Γ( − z ) ζ ∗ (1 + 2 z ) × (cid:34) h r − z Γ( − z + r )Γ(2 z − r ) e ( + z − r − s ) Γ( k − + z − r )( + z − r − s ) + h r Γ( + z + r )Γ( − z − r ) e ( − z − r − s ) Γ( k − − z − r )( − z − r − s ) (cid:35) dz + Ω( s ; h, δ ) − O s,c ( δ )where C σ is as in (4.12). Combining (5.7) and (5.11) we get that D ( s ; h, δ )= 12 πi (cid:90) (2) D ( u ; h, δ ) e ( u − s ) duu − s (5.12a) − πi (cid:90) ( c ) ( D ( u ; h, δ ) − Ω( u ; h, δ )) e ( u − s ) duu − s (5.12b) − (cid:98) − c (cid:99) (cid:88) r =0 (cid:88) a πi (cid:90) C σ V π − z ρ a ( − z, − h ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) − r r !(4 π ) − k Γ( + z )Γ( − z ) ζ ∗ (1 + 2 z ) × (cid:34) h r − z Γ( − z + r )Γ(2 z − r ) e ( + z − r − s ) Γ( k − + z − r )( + z − r − s ) + h r Γ( + z + r )Γ( − z − r ) e ( − z − r − s ) Γ( k − − z − r )( − z − r − s ) (cid:35) dz + Ω( s ; h, δ ) − (cid:88) t j , ≤ r< ∓ Im ( t j ) − c R ( + it j − r ; h, δ ) e ( + it j − r − s ) + it j − r − s By (4.14), we see that when Re ( u ) > 1, the limit as δ → D ( σ + it ; h ) for σ > e u , the limit as δ → t , and Re s ≤ + | Im t | ,lim δ → M ( s, t, δ ) exists and M ( s, t ) = lim δ → M ( s, t, δ ) = √ π − s Γ( s − − it )Γ( s − + it )Γ(1 − s )Γ( − it )Γ( + it ) . (5.13) ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 33 Referring to Proposition 4.4, line (4.24), it is clear that in strip − A + (cid:15) < σ < − k the limit as δ → D ( s ; h, δ ) exists, equals D ( s ; h ) as defined by D ( s ; h ) = (4 π ) k s + k − h s − (cid:32)(cid:88) j ρ j ( − h ) Γ( s − − it j )Γ( s − + it j )Γ(1 − s )Γ( − it j )Γ( + it j ) (cid:104) V, u j (cid:105) + (cid:88) a (cid:34) πi (cid:90) − C σ V π − z | h | − z ρ a (cid:0) − z, − h (cid:1) Γ( − z ) ζ ∗ (1 + 2 z ) × Γ( s − − z )Γ( s − + z )Γ(1 − s )Γ( − z )Γ( + z ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) dz (cid:21)(cid:19) + Ω( s ; h ) , (5.14)whereΩ( s ; h ) = (4 π ) k s + k − (cid:88) a (cid:98) − σ (cid:99) (cid:88) (cid:96) =0 ( − (cid:96) Γ(1 − s )Γ(2 s + (cid:96) − (cid:96) !Γ( s + (cid:96) )Γ(1 − s − (cid:96) ) (5.15) × (cid:20) π − s − (cid:96) h − s − (cid:96) V ρ a (1 − s − (cid:96), − h )Γ(1 − s − (cid:96) ) ζ ∗ (2 s + 2 (cid:96) ) (cid:104) V, E ∗ a ( ∗ , s + (cid:96) ) (cid:105) +(1 − δδδ σ,(cid:96) ) π s + (cid:96) h (cid:96) V ρ a ( s + (cid:96), − h )Γ ( s + (cid:96) ) ζ ∗ (2 − s − (cid:96) ) (cid:104) V, E ∗ a ( ∗ , − s − (cid:96) ) (cid:105) (cid:21) , and where δδδ σ,(cid:96) is given in (4.13).Furthermore, in this range D ( s ; h ) satisfies the upper bound D ( s ; h ) − Ω( s ; h ) (cid:28) A,(cid:15) (1 + | s | ) A (cid:48) ( (cid:96) (cid:96) ) (1 − k ) / h − σ × max (cid:32) , ( h, (cid:96) (cid:96) ) / σ ( (cid:96) (cid:96) ) h / log(3 h )( (cid:96) (cid:96) ) (cid:33) , (5.16)where | Im s | is sufficiently large, A (cid:48) is determined by A and s is at least (cid:15) away from thepoles at ± it j − r . Also, when ( h, N ) = 1, the term max ( ∗ ) can be replaced by h θ , where θ represents the best progress toward the Ramanujan conjecture. The exponential decayof D ( s ; h ) in the imaginary part of s shown in (5.16), together with the square-exponentialdecay of e u shows that for − A + (cid:15) < Re s < − k , the limit as δ → R ( + it j − r ; h, δ ) grows atmost polynomially in | t j | for any fixed s with − A + (cid:15) < Re s < 2. As e ( + it j − r − s ) decays square-exponentially in | t j | , the sum over the residues converges and approachesthe limits given above in (5.2) and (5.17). We have thus established that for any s inthe range − A + (cid:15) < Re s < 2, the limit as δ → D ( s ; h, δ ) exists. Indeed, for s inthe vertical strip c < σ < 2, at least (cid:15) away from the poles noted above for any c with − A + (cid:15) < c < − k then the limit, D ( s ; h ), is given by D ( s ; h ) = (5.17)12 πi (cid:90) i ∞ − i ∞ D ( u ; h ) e ( u − s ) duu − s − πi (cid:90) c + i ∞ c − i ∞ ( D ( u ; h ) − Ω( u ; h )) e ( u − s ) duu − s − (cid:98) − c (cid:99) (cid:88) r =0 (cid:88) a πi (cid:90) C σ V π − z ρ a ( − z, − h ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) − r r !(4 π ) − k Γ( + z )Γ( − z ) ζ ∗ (1 + 2 z ) × (cid:34) h r − z Γ( − z + r )Γ(2 z − r ) e ( + z − r − s ) Γ( k − + z − r )( + z − r − s ) + h r Γ( + z + r )Γ( − z − r ) e ( − z − r − s ) Γ( k − − z − r )( − z − r − s ) (cid:35) dz + Ω( s ; h ) − (cid:88) ± t j , ≤ r< ∓ Im ( t j ) − c R ( + it j − r ; h ) e ( + it j − r − s ) + it j − r − s (5.18)Here in the integral with Re u = 2, D ( u ; h ) is given by (5.1), and in the integral with Re u = c , D ( u ; h ) is given by (5.14). In this strip, D ( s ; h, δ ) satisfies the same upperbound as given in (5.16).The average upper bounds for the c r,j are obtained by an application of Stirling’s formula,and Propositions 4.1 and 4.3. It follows then that (cid:88) t j , ≤ r< − c R ( + it j − r ; h ) e ( + it j − r − s ) + it j − r − s (cid:28) c (log( (cid:96) (cid:96) )) ( (cid:96) (cid:96) ) − k h − σ + θ . (5.19)By (5.16), the integral along Re s = c is (cid:28) (1 + | s | ) A (cid:48) ( (cid:96) (cid:96) ) − k h − c max (cid:32) h (cid:15) , ( h, (cid:96) (cid:96) ) / σ ( (cid:96) (cid:96) ) h / log(3 h )( (cid:96) (cid:96) ) (cid:33) , (5.20)and by (4.25), the integral along Re s = 2 is (cid:28) (cid:16) h(cid:96) (cid:17) ( k − / (cid:96) ( (cid:96) (cid:96) ) ( k − / (1 + | s | ) A (cid:48) (cid:28) (cid:96) − h ( k − / ( (cid:96) (cid:96) ) − ( k − / (1 + | s | ) A (cid:48) . Noting that − c > k − , it follows from the previous three lines that for any s with c < Re s < D ( s ; h ) − Ω( s ; h ) satisfies the upper bound (5.20).When t j = 0 for some j , the above argument is nearly identical except for additional,slightly different residual terms from (5.5) arising from the double poles of D ( s ; h, δ ),indicating that D ( s ; h ) has double poles at s = − r . (cid:3) ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 35 An application to single shifted sums We begin by defining the smoothed shifted sum, S ( x ; h ) := (cid:88) (cid:96) m = (cid:96) m + h A ( m ) B ( m ) G ( m /x ) . Here G ( x ) = 12 πi (cid:90) i ∞ − i ∞ g ( s ) x s ds (6.1)is smooth, with compact support in the interval [1 , 2] and g ( s ) is holomorphic, witharbitrary polynomial decay in vertical strips. The objective of this section is to prove thefollowing theorem. Theorem 6.1. Fix N , (cid:96) , (cid:96) , h ∈ N , with N , (cid:96) , (cid:96) square free and ( N , (cid:96) (cid:96) ) = 1 . Suppose h (cid:28) (cid:96) x . Then if θ > , S ( x ; h ) (cid:28) ( (cid:96) x ) ( (cid:96) x/h ) θ max (cid:32) , ( h, (cid:96) (cid:96) ) / σ ( (cid:96) (cid:96) ) h / log(3 h )( (cid:96) (cid:96) ) (cid:33) . Also if ( N (cid:96) (cid:96) , h ) = 1 , then S ( x ; h ) (cid:28) ( (cid:96) x ) ( (cid:96) x/h ) θ h θ = ( (cid:96) x ) + θ . The above bounds still hold if θ = 0 , unless t j = 0 for some j , in which case we replace ( (cid:96) x/h ) θ with log( (cid:96) x/h ) .Suppose h (cid:29) (cid:96) x , then S ( x ; h ) (cid:28) h ( h/(cid:96) x ) (cid:15) max (cid:32) , ( h, (cid:96) (cid:96) ) / σ ( (cid:96) (cid:96) ) h / log(3 h )( (cid:96) (cid:96) ) (cid:33) . (6.2) Also, if ( N (cid:96) (cid:96) , h ) = 1 then S ( x ; h ) (cid:28) h + θ ( h/(cid:96) x ) (cid:15) . (6.3)The remainder of this section is devoted to the proof of this result. To the best of theknowledge of the authors, this theorem roughly equals the current state of the art inestimates for single shifted sums, although the proof is accomplished by new methods.We include this mainly as a reference point from which we will build a new theory ofdouble shifted sums.6.1. The expression of the shifted sum as an inverse Mellin transform. Thesmoothed sum S ( x ; h ) can be expressed as an inverse Mellin transform of D ( s ; h ) asfollows. Proposition 6.2. For < γ < ( k − / , S ( x ; h ) = ( (cid:96) (cid:96) ) ( k − / (2 πi ) (cid:90) ( γ ) (cid:90) (2) Γ( u )Γ(( k − / − u ) D ( s ; h )( (cid:96) x ) s + u g ( s + u ) dsdu Γ(( k − / h u . Proof. We observe that, S ( x ; h ) = (cid:88) (cid:96) m = (cid:96) m + h πi (cid:90) (2) a ( m )¯ b ( m ) g ( s ) x s dsm ( k − / m ( k − / s = (cid:88) (cid:96) m = (cid:96) m + h ( (cid:96) (cid:96) ) ( k − / πi (cid:90) (2) a ( m )¯ b ( m ) g ( s )( (cid:96) m ) ( k − / ( (cid:96) m ) ( k − / (cid:18) (cid:96) x(cid:96) m (cid:19) s ds = (cid:88) (cid:96) m = (cid:96) m + h ( (cid:96) (cid:96) ) ( k − / πi (cid:90) (2) a ( m )¯ b ( m )( (cid:96) x ) s g ( s ) ds ( (cid:96) m + h ) ( k − / ( (cid:96) m ) ( k − / s = (cid:88) (cid:96) m = (cid:96) m + h ( (cid:96) (cid:96) ) ( k − / πi (cid:90) (2) a ( m )¯ b ( m )( (cid:96) m ) k − s (cid:18) h(cid:96) m (cid:19) − ( k − / ( (cid:96) x ) s g ( s ) ds. A form of the following identity can be found in [20], 6.422(3). For Re β > γ > t ) < π , 12 πi (cid:90) ( γ ) Γ( u )Γ( β − u ) t − u du = Γ( β )(1 + t ) − β . (6.4)Move the line of integration to Re s = 2 + γ . Then setting β = ( k − / t = h/ ( (cid:96) m )and substituting into the above, we have S ( x ; h ) = (cid:88) (cid:96) m = (cid:96) m + h ( (cid:96) (cid:96) ) ( k − / (2 πi ) (cid:90) (2+ γ ) (cid:90) ( γ ) a ( m )¯ b ( m )Γ( u )Γ(( k − / − u )( (cid:96) m ) k − s − u Γ(( k − / h u ( (cid:96) x ) s g ( s ) duds = ( (cid:96) (cid:96) ) ( k − / (2 πi ) (cid:90) (2+ γ ) (cid:90) ( γ ) Γ( u )Γ(( k − / − u ) D ( s − u ; h )( (cid:96) x ) s g ( s ) duds Γ(( k − / h u = ( (cid:96) (cid:96) ) ( k − / (2 πi ) (cid:90) (2) (cid:90) ( γ ) Γ( u )Γ(( k − / − u ) D ( s ; h )( (cid:96) x ) s + u g ( s + u ) duds Γ(( k − / h u The proposition follows after interchanging the order of the integrals. (cid:3) Moving the line of integration. Taking the expression from Proposition 6.2, wemove the s line of integration to the contour, C , just to the left of the line Re s = − k butstill to the right of the zeros of ζ (2 s + k ). We bend C so that it passes over every pole ofthe form + it j − r , including those potential exceptional poles to the left of Re s = − k that would lie on the real line. ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 37 Thus by Proposition 5.1, if we assume that t j (cid:54) = 0 for all j then we obtain S ( x ; h ) (6.5)= ( (cid:96) (cid:96) ) ( k − / (2 πi ) (cid:90) ( γ ) (cid:90) C Γ( u )Γ(( k − / − u )( D ( s ; h ) − Ω( s ; h ))( (cid:96) x ) s + u g ( s + u ) dsdu Γ(( k − / h u + (cid:88) j, ≤ r< − c ( (cid:96) (cid:96) ) ( k − / πi (cid:90) ( γ ) Γ( u )Γ(( k − / − u ) c r,j ρ j ( − h )( (cid:96) x ) g ( + u − r + it j ) du Γ(( k − / h/(cid:96) x ) u − r + it j + ( (cid:96) (cid:96) ) ( k − / (2 πi ) (cid:90) ( γ ) (cid:90) ( ) Γ( u )Γ(( k − / − u )Ω( s ; h )( (cid:96) x ) s + u g ( s + u ) dsdu Γ(( k − / h u , where the last integral above is more of a shorthand for the residual terms due to thepolar contributors from the continuous part of D ( s ; h ), since Ω( s ; h ) is defined piecewisealong vertical strips,. Indeed, if we expand the last integral we get that( (cid:96) (cid:96) ) ( k − / (2 πi ) (cid:90) ( γ ) (cid:90) ( ) Γ( u )Γ(( k − / − u )Ω( s ; h )( (cid:96) x ) s + u g ( s + u ) dsdu Γ(( k − / h u = (cid:88) a k (cid:88) r =0 (4 π ) k ( − r ( (cid:96) (cid:96) ) ( k − / r !(2 πi ) (cid:90) ( γ ) (cid:90) ( − r ) Γ( u )Γ(( k − / − u )( (cid:96) x ) s + u g ( s + u )Γ(( k − / s + k − h u × Γ(1 − s )Γ(2 s + r − s + r )Γ(1 − s − r ) (cid:20) π − s − r h − s − r V ρ a (1 − s − r, − h )Γ(1 − s − r ) ζ ∗ (2 s + 2 r ) (cid:104) V, E ∗ a ( ∗ , s + r ) (cid:105) + π s + r h (cid:96) V ρ a ( s + r, − h )Γ ( s + r ) ζ ∗ (2 − s − r ) (cid:104) V, E ∗ a ( ∗ , − s − r ) (cid:105) (cid:21) dsdu. (6.6)Changing s → s + − r and changing the order of integration we get that this becomes= (cid:88) a , ≤ r ≤ k (4 π ) k ( − r ( (cid:96) (cid:96) ) ( k − / r !(2 πi ) (cid:90) (cid:90) (0) ( γ ) Γ( u )Γ(( k − / − u )( (cid:96) x ) − r + s + u g ( − r + s + u )Γ(( k − / s + k − − r ) h s + u − r × Γ( − r − s )Γ(2 s − r )Γ( s + )Γ( − s ) (cid:34) π − s h − s V ρ a (cid:0) − s, − h (cid:1) Γ( − s ) ζ ∗ (1 + 2 s ) (cid:104) V, E ∗ a (cid:0) ∗ , + s (cid:1) (cid:105) + π + s h s V ρ a (cid:0) + s, − h (cid:1) Γ (cid:0) + s (cid:1) ζ ∗ (1 − s ) (cid:104) V, E ∗ a ( ∗ , − s ) (cid:105) (cid:35) duds, (6.7)and so we see this is actually completely analogous to the contribution from the poles at s = + it j − r from the discrete part of the spectrum in (6.5). Since D ( s ; h ) − Ω( s ; h ) satisfies the upper bound given in (5.16) when σ < − k , we seethe size of the first double integral in (6.5) is controlled by the factor h (cid:18) (cid:96) xh (cid:19) s + u max (cid:32) , ( h, (cid:96) (cid:96) ) / σ ( (cid:96) (cid:96) ) h / log(3 h )( (cid:96) (cid:96) ) (cid:33) or furthermore, when ( h, N ) = 1, by h + θ (cid:18) (cid:96) xh (cid:19) s + u . Similarly, we use Proposition 4.3, the bound in (4.5), and the expansion in (6.7) to showthat the residual terms and the integral of Ω( h, s ) are bounded by( (cid:96) x ) (cid:18) (cid:96) xh (cid:19) Re ( it j + u ) max (cid:32) , σ ( (cid:96) (cid:96) ) (cid:18) h ( h, (cid:96) (cid:96) )( (cid:96) (cid:96) ) (cid:19) / log(3 h ) (cid:33) or alternately, when ( h, N ) = 1, ( (cid:96) x ) (cid:18) (cid:96) xh (cid:19) Re ( it j + u ) h θ . Thus in both cases, it is optimal to make the exponent as small as possible if (cid:96) x > h ,and as large as possible if (cid:96) x < h . If (cid:96) x = h then it doesn’t matter.In the case (cid:96) x > h , we take the integral component and we lower u past the pole at 0,to − (cid:15) . Thus for θ > (cid:96) x ) ( (cid:96) x/h ) θ max (cid:32) , ( h, (cid:96) (cid:96) ) / σ ( (cid:96) (cid:96) ) h / log(3 h )( (cid:96) (cid:96) ) (cid:33) . In particular, if (cid:96) , (cid:96) are prime and relatively prime to h then S ( x ; h ) (cid:28) ( (cid:96) x ) ( (cid:96) x/h ) θ max (cid:32) , h / log(3 h )( (cid:96) (cid:96) ) (cid:33) , (6.8)and alternatively, if ( N, h ) = 1 then S ( x ; h ) (cid:28) ( (cid:96) x ) ( (cid:96) x/h ) θ h θ = ( (cid:96) x ) + θ . The case where t j = 0 for some j only alters the above argument when θ = 0, as it willcontribute the main term. In particular, it adds terms to the residual spectrum of theform log( (cid:96) x/h ) rather than the ( (cid:96) x/h ) θ that appears in (6.8).In the case (cid:96) x < h , we get the greatest contribution from the smallest exponent. Con-sidering at the factor h (cid:18) (cid:96) xh (cid:19) s + u , (6.9) ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 39 we increase u past the pole at ( k − / 2, to ( k + 1) / − (cid:15) . In this case the dominantcontribution comes from the first double integral term in (6.5). Since σ is always justslightly to the left of − k we obtain the bound S ( x ; h ) (cid:28) h ( h/(cid:96) x ) (cid:15) max (cid:32) , h / ( (cid:96) (cid:96) ) (cid:33) . (6.10)Alternatively, if ( N, h ) = 1 then S ( x ; h ) (cid:28) h + θ ( h/(cid:96) x ) (cid:15) . (6.11)This completes the proof of the Theorem 6.1. (cid:3) A double Dirichlet series Recall that D ( s ; h ) = (cid:88) m (cid:96) = m (cid:96) + h a ( m )¯ b ( m )( m (cid:96) ) s + k − (7.1)is convergent when Re s > N , (cid:96) , (cid:96) ∈ N , with N , (cid:96) , (cid:96) square free, ( N , (cid:96) (cid:96) ) = 1 and have set N = N (cid:96) (cid:96) / ( (cid:96) , (cid:96) ). Fix a positive integer Q , with ( Q, N ) = 1. For s, w with real partsgreater than one, the sum Z Q ( s, w ) = ( (cid:96) (cid:96) ) ( k − / (cid:88) h ≥ D ( s ; h Q )( h Q ) w +( k − / (7.2)is convergent, and can also be written in the form Z Q ( s, w ) = (cid:88) h ≥ (cid:96) m (cid:96) m h Q A ( m ) ¯ B ( m ) (cid:16) h Q(cid:96) m (cid:17) ( k − / ( (cid:96) m ) s ( h Q ) w +( k − / . (7.3)The object of this section is to obtain the meromorphic continuation of Z Q ( s, w ) to thepart of C with Re w > 1. The function Z Q ( s, w ) can be meromorphically continued toall of C , but as this lengthens the argument, and we have no immediate application, wewill instead refer to [25], where this second continuation is obtained in a different way.Before stating our first proposition we need to make several definitions. For Re s (cid:48) > L Q ( s (cid:48) , u j ) = (cid:88) h ≥ ρ j ( − h Q )( h Q ) s (cid:48) . (7.4)This is, up to the presence of the factor Q , the familiar L -series associated to u j , but withcoefficients normalized so that the inner product of u j with itself equals one. We also define the function ζ a ,Q ( s (cid:48) , z ), for each cusp a , by a definition analogous to L Q ,with ζ a ,Q ( s (cid:48) , z ) = Q − z ζ (1 − z ) (cid:88) h ≥ ρ a (cid:0) − z, − hQ (cid:1) h s (cid:48) + z . Substituting (2.15) and summing, we obtain the following, as in [25] line (2.34). By ourassumptions, N is square free. For each cusp a = w for w | N , we have, for Q ≥ ζ a ,Q ( s (cid:48) , z ) = ζ ( s (cid:48) + z ) ζ ( s (cid:48) − z ) (cid:32) Nw (cid:33) − z Q − z (cid:89) p | N (cid:0) − p − z (cid:1) − (cid:89) p | Nw (1 − p − ( s (cid:48) − z ) ) × (cid:89) p | w,pα (cid:107) Q,α ≥ p − z − p z (cid:16) ( p − − p − ( s (cid:48) − z ) ) + p α z ( p z − p )(1 − p − ( s (cid:48) + z ) ) (cid:17) × (cid:89) q | Q,q (cid:45) N,qα (cid:107) Q (cid:0) − q z (cid:1) − (cid:16) (1 − q − ( s (cid:48) − z ) ) − q ( α +1)2 z (1 − q − ( s (cid:48) + z ) ) (cid:17) . (7.5)Finally, we define a ratio of Dirichlet polynomials K ± a ,Q ( s (cid:48) ) by settingRes z = ± (1 − s (cid:48) ) ζ a ,Q ( s (cid:48) , z ) = K ± a ,Q ( s (cid:48) ) ζ ( − s (cid:48) ) . (7.6)With this notation in place, we state the following Proposition 7.1. Choose c , with − A < c < − k . Let (cid:96) , (cid:96) ∼ L as in Theorem 1.3.The function Z Q ( s, w ) continues to the region Re s > c , Re w > and in this region, itis meromorphic. If t j (cid:54) = 0 for all Maass forms at level N , then Z Q ( s, w ) has simple polesat the points s = − r ± it j , for ≤ r < − c ∓ Im ( t j ) . Furthermore it has poles due to Ψ( s, w ) as defined in (7.31) , which are simple poles when s = − r for r ≥ and poleswhich are in the same place as the poles of ζ (2 s + 2 r ) − for r ≥ . There are also simplepolar lines at w + 2 s + k = − (cid:96) , for integral (cid:96) ≥ . If there is a Maass form at this levelwith t j = 0 , then there are double poles at s = − r for A > r ≥ .When s = − r + it j , for t j (cid:54) = 0 , we have Res s = − r + it j Z Q ( s, w ) = ( (cid:96) (cid:96) ) ( k − / c r,j L Q ( w + ( k − / − r + it j , u j ) , where the index j differentiates between + it j and − it j and c r,j is given by (5.3) . ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 41 When w + 2 s + k = − (cid:96) Res w +2 s + k = − (cid:96) Z Q ( s, w ) == (4 π ) k Γ(1 − s )2Γ( s + k − (cid:88) a (cid:34) V ( − (cid:96) Γ(2 s + (cid:96) − Q − ( − s − (cid:96) ) (cid:96) ! π s + (cid:96) − ζ ∗ (2 s + 2 (cid:96) )Γ( s + (cid:96) )Γ(1 − s − (cid:96) ) (7.7) × (cid:16) K + a ,Q ( − s − (cid:96) ) (cid:104) V, E ∗ a ( ∗ , s + (cid:96) ) (cid:105) + K − a ,Q ( − s − (cid:96) ) (cid:104) V, E ∗ a ( ∗ , − s − (cid:96) ) (cid:105) (cid:17)(cid:105) . Set s (cid:48) = s + w + k − and fix (cid:15) > . For − A < Re s < − k and Re w > , we have,away from the poles, the convergent spectral expansion Z Q ( s, w ) = ( (cid:96) (cid:96) ) ( k − / ( S cusp ( s, s (cid:48) ) + S cts,int ( s, s (cid:48) )) + Ψ( s, w ) , where S cusp ( s, s (cid:48) ) := (4 π ) k Γ(1 − s )2Γ( s + k − (cid:88) j L Q ( s (cid:48) , u j )Γ( s − − it j )Γ( s − + it j ) (cid:104) V, u j (cid:105) Γ( − it j )Γ( + it j ) , (7.8) S cts,int ( s, s (cid:48) ) is given by S cts,int = (4 π ) k Γ(1 − s )2Γ( s + k − × (cid:88) a πi (cid:90) − C σ,σ (cid:48) V Q − s (cid:48) ζ a ,Q ( s (cid:48) , z )Γ( s − − z )Γ( s − + z ) ζ ∗ (1 + 2 z ) ζ ∗ (1 − z )Γ( − z )Γ( + z ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) dz, (7.9) and the curve of integration is described in (7.33) . We also have the following upperbounds: For Re s (cid:48) ≥ , s at least (cid:15) away from the poles, and Re w > , there exists aconstant A (cid:48)(cid:48) , depending only on k and A , such that Z Q ( s, w ) − Ψ( s, w ) (cid:28) Q θ − Re ( s (cid:48) )+ (cid:15) (1 + | s | ) A (cid:48)(cid:48) (1 + | s (cid:48) | ) (cid:15) . (7.10) For Re s (cid:48) < and Re w > , Z Q ( s, w ) − Ψ( s, w ) (7.11) (cid:28) (cid:96) ,(cid:96) Q − Re ( s (cid:48) )+ (cid:15) (cid:89) p γ || Q (cid:12)(cid:12)(cid:12) λ j ( p γ ) − λ j ( p γ − ) p − s (cid:48) (cid:12)(cid:12)(cid:12) (1 + | s | ) A (cid:48) (1 + | s (cid:48) | ) − Re ( s (cid:48) ) . In this last case, the dependence on (cid:96) , (cid:96) is suppressed, as our applications do not requirea precise estimate in this region.Proof. Applying Proposition 5.1, in particular the upper bound (5.20) and the formulationfor Ω( s ; h ) in (5.15), we see that the summation (7.2) is locally normally convergent, for w with sufficiently large real part, when c < Re s < 2. For s in the range Re s < − k − (cid:15) ,each D ( s ; h Q ) can be represented by a convergent spectral sum, when s is a distanceof at least (cid:15) from any pole. Applying Proposition 5.1, when s is in this range we can interchange this sum over t j with the sum over h and obtain, for Re w sufficiently large,a spectral expansion for Z Q ( s, w ). Writing s (cid:48) = w + s + k − Z Q ( s, w ) = ( (cid:96) (cid:96) ) ( k − / ( S cusp ( s, s (cid:48) ) + S cts ( s, s (cid:48) )) , (7.12)where S cusp ( s, s (cid:48) ) is given by (7.8), and S cts ( s, s (cid:48) ) := (4 π ) k Γ(1 − s )2Γ( s + k − (cid:88) a (cid:20) πi (cid:90) − C σ V Q − s (cid:48) ζ a ,Q ( s (cid:48) , z ) ζ ∗ (1 − z ) ζ ∗ (1 + 2 z ) × Γ( s − − z )Γ( s − + z )Γ( − z )Γ( + z ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) dz + (cid:98) − σ (cid:99) (cid:88) (cid:96) =0 (cid:32) V Q − s (cid:48) ζ a ,Q (cid:0) s (cid:48) , − s − (cid:96) (cid:1) ζ ∗ (2 − s − (cid:96) ) ζ ∗ (2 s + 2 (cid:96) ) (cid:104) V, E ∗ a ( ∗ , s + (cid:96) ) (cid:105) + (1 − δδδ σ,(cid:96) ) V Q − s (cid:48) ζ a ,Q (cid:0) s (cid:48) , − + s + (cid:96) (cid:1) ζ ∗ (2 s + 2 (cid:96) ) ζ ∗ (2 − s − (cid:96) ) (cid:104) V, E ∗ a ( ∗ , − s − (cid:96) ) (cid:105) (cid:33) × ( − (cid:96) Γ(2 s + (cid:96) − (cid:96) !Γ( s + (cid:96) )Γ(1 − s − (cid:96) ) (cid:21) . (7.13)Here C σ and δδδ σ,(cid:96) are as described in (4.12) and (4.13). This sum has been taken in theregion where the zeta function converges, where Re s (cid:48) > 1. We will now extend the regionin which Z Q ( s, w ) is meromorphic to the region Re w > Re s < − k , with Re s (cid:48) ≤ (cid:88) | t j |∼ T L Q ( s (cid:48) , u j ) (cid:104) V, u j (cid:105) = (cid:88) | t j |∼ T (cid:88) h ≥ ρ j ( − h Q ) (cid:104) V, u j (cid:105) ( h Q ) s (cid:48) . = (cid:88) | t j |∼ T L ( s (cid:48) , u j ) (cid:104) V, u j (cid:105) Q − s (cid:48) (cid:89) p γ || Q (cid:16) λ j ( p γ ) − λ j ( p γ − ) p − s (cid:48) (cid:17) . (7.14)Note that as Q has been chosen to be relatively prime to N , we may assume that, forboth oldforms and newforms, the u j are eigenfunctions of the Hecke operators at primes p dividing Q , with eigenvalues λ j ( p ). A consequence of this is that for ( n, Q ) = 1, and p dividing Q , ρ j ( np γ ) = ρ j ( n ) λ j ( p γ ) . The right hand side of (7.14) then follows after a simple computation.Suppose that Re s (cid:48) < 0. If u j is a newform then by the functional equation we have thebound for a single L ( s (cid:48) , u j ), which we denote simply by L ( s (cid:48) , u j ). L ( s (cid:48) , u j ) (cid:28) (1 + | s (cid:48) | + | t j | ) − Re s (cid:48) ( (cid:96) (cid:96) ) − Re s (cid:48) | L (1 − s (cid:48) , u j ) | . ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 43 If u j is an oldform then the dependence on (cid:96) (cid:96) is slightly more complicated. As we donot need precise dependence on (cid:96) (cid:96) for our applications, we will suppress the dependenceon (cid:96) (cid:96) and, for Re s (cid:48) < 0, will use the simpler bound, L Q ( s (cid:48) , u j ) (cid:28) (cid:96) ,(cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q − Re s (cid:48) (cid:89) p γ || Q (cid:16) λ j ( p γ ) − λ j ( p γ − ) p − Re s (cid:48) (cid:17) (1 + | s (cid:48) | + | t j | ) − Re s (cid:48) L (1 − s (cid:48) , u j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (7.15)For Re s (cid:48) ≥ , we apply Cauchy-Schwarz to the left hand side of (7.14), obtaining (cid:88) | t j |∼ T L Q ( s (cid:48) , u j ) (cid:104) V, u j (cid:105) ≤ (cid:88) | t j |∼ T | L Q ( s (cid:48) , u j ) | e − π | t j | (cid:88) | t j |∼ T | e π | t j | (cid:104) V, u j (cid:105) | From Proposition 4.1 we have (cid:88) | t j |∼ T e π | t j | |(cid:104) V, u j (cid:105)| (cid:28) L − k T k log T. (7.16)For Re s (cid:48) ≥ , the L -series average is addressed by the bound, (cid:88) | t j |∼ T e − π | t j | | L Q ( s (cid:48) , u j ) | (cid:28) Q θ − Re s (cid:48) + (cid:15) L (cid:15) (1 + | s (cid:48) | + T ) (cid:15) . (7.17) Remark 7.2. By (7.14), L Q ( s (cid:48) , u j ) ≈ λ j ( Q ) Q − s (cid:48) L ( s (cid:48) , u j ) . To obtain (7.17) we have simply estimated | λ j ( Q ) | from above, using the best progresstoward the Ramanujan Conjecture, and applied a simple second moment estimate for the L -series part that is uniform in the level.Combining (7.16) with (7.17) gives us the desired result. For Re s (cid:48) ≥ , (cid:88) | t j |∼ T L Q ( s (cid:48) , u j ) (cid:104) V, u j (cid:105) (cid:28) Q θ − s (cid:48) + (cid:15) L − k + (cid:15) (1 + | γ (cid:48) | + | T | ) k + (cid:15) . (7.18)To understand the region of absolute convergence we now rewrite (7.8) as S cusp = lim T →∞ S cusp ( T ) , with S cusp ( T ) = 2 k − π k Γ(1 − s )Γ( s + k − 1) (7.19) × (cid:88) | t j |(cid:28) T L Q ( s (cid:48) , u j )Γ( s − − it j )Γ( s − + it j ) (cid:104) V, u j (cid:105) Γ( − it j )Γ( + it j ) The sum S cusp ( T ) can in turn be rewritten, after breaking up the interval [0 , T ] dyadically,as S cusp ( T ) (cid:28) ∞ (cid:88) i =0 S (cid:48) cusp ( T / i )where S (cid:48) cusp ( T ) is the same as S cusp ( T ), but with the condition | t j | (cid:28) T replaced by | t j | ∼ T . By Stirling’s formula we haveΓ(1 − s )Γ( s − − it j )Γ( s − + it j )Γ( s + k − − it j )Γ( + it j ) (cid:28) (1 + | t j | ) σ − (1 + | s | ) − k − σ , (7.20)which is valid regardless of the relative sizes of t j and Im s for s at least (cid:15) away frompoles.We now substitute the upper bounds of (7.15), (7.18) and (7.20), obtaining, for Re s (cid:48) < S (cid:48) cusp ( T ) (7.21) (cid:28) (cid:96) ,(cid:96) Q max( θ − Re s (cid:48) , − Re s (cid:48) )+ (cid:15) (1 + | s (cid:48) | + T ) − Re s (cid:48) + (cid:15) T σ + k − (1 + | s | ) − k − σ . In the case Re s (cid:48) ≥ , we similarly get the more precise bound( (cid:96) (cid:96) ) ( k − / S (cid:48) cusp ( T ) (7.22) (cid:28) Q θ − Re s (cid:48) + (cid:15) (1 + | s (cid:48) | + T ) (cid:15) T σ + k − (1 + | s | ) − k − σ . Note that there may be exceptional poles on the real line, close to s = 1 / − k/ 2. Aslong as s is at least (cid:15) away from any of these, the contribution from this piece of the sumhas already been accounted for in the estimate. The interval 0 ≤ Re s (cid:48) ≤ is filled in byconvexity. Convergence of the expansion given in (7.8) as T → ∞ , away from the poles,will occur when the exponent of T in (7.21) and (7.22) is negative. This occurs for Re s < (cid:40) Re s (cid:48) − k if Re s (cid:48) < , − k if Re s (cid:48) ≥ . (7.23)As s (cid:48) = s + w + k − 1, this implies that S cusp ( s, s (cid:48) ) is convergent and analytic in s and s (cid:48) away from poles when Re s < − k and Re w > 1, and satisfies S cusp ( s, s (cid:48) ) (cid:28) (cid:96) ,(cid:96) Q θ − Re s (cid:48) (1 + | s (cid:48) | ) (cid:15) (1 + | s | ) − k − σ . (7.24)7.1. The continuous piece. We now turn to S cts ( s, s (cid:48) ). This piece converges in thesame region as S cusp ( s, s (cid:48) ) and satisfies a stronger upper bound. To demonstrate this weneed to estimate (cid:88) a πi (cid:90) (0) V Q − s (cid:48) ζ a ,Q ( s (cid:48) , z ) ζ ∗ (1 − z ) ζ ∗ (1 + 2 z ) Γ( s − − z )Γ( s − + z )Γ( − z )Γ( + z ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) dz. ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 45 Recall again that N is square-free. As f and g are new forms, by [1], applying Lemma5.7 in [25], and summing over the cusps, we see that √V (cid:88) a (cid:10) V, E ∗ a (cid:0) ∗ , + z (cid:1)(cid:11) (cid:28) ζ ∗ (1 + 2 z ) L − k − (1 + | t | ) k − e − π | t | / L ( + it, f ⊗ g ) log(1 + | t | ) N (cid:15) . (7.25)Applying the convexity bound to L ( + it, f ⊗ g ), Stirling’s formula, (7.5), and the factthat (cid:90) T ζ ( s (cid:48) + it ) ζ ( s (cid:48) − it ) dt (cid:28) s (cid:48) T (cid:15) (7.26)when Re s (cid:48) ≥ with at most polynomial dependence on s (cid:48) , we obtain convergence in thedesired region.In this region we observe that S cts ( s, s (cid:48) ) has poles due the continuous part of D ( s ; h ) at s = − r for r ≥ 1, and when s = (cid:37) − r with r ≥ (cid:37) a nontrivial zero of ζ . We alsoobserve that there are new potential poles in this region due to ζ ( s (cid:48) − + s + (cid:96) ), where (cid:96) ≤ (cid:98) − σ (cid:99) . There are no poles due to ζ ( s (cid:48) + − s − (cid:96) ) in this region since − Re s > (cid:96) − so Re s (cid:48) + − Re s − (cid:96) > Re s at least (cid:15) > − r for r ≥ 0, then for Re s (cid:48) > 1, theintegral component of the continuous contribution in (7.13) is, for Re w sufficiently largeand Re s < − k , S int ( s, s (cid:48) ) := (4 π ) k Γ(1 − s )2Γ( s + k − 1) (7.27) × (cid:88) a (cid:32) πi (cid:90) (0) V Q − s (cid:48) ζ a ,Q ( s (cid:48) , z )Γ( s − − z )Γ( s − + z ) ζ ∗ (1 + 2 z ) ζ ∗ (1 − z )Γ( − z )Γ( + z ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) dz (cid:33) . This converges when Re s (cid:48) > 1, as the zeta factors are (cid:28) Re s (cid:48) ≤ 1, we take s (cid:48) to be within theregion to the right of Re s (cid:48) = 1 and to the left of 1 + C , where C ( x ) is as in description of(4.10). Without loss of generality, we can also choose (cid:15) > (cid:15) > | Re C ( x ) | > x . Shifting the line of integration to − C , a pole of ζ a ,Q ( s (cid:48) , z ) is passed over at z = 1 − s (cid:48) , giving S int ( s, s (cid:48) ) = (4 π ) k Γ(1 − s )2Γ( s + k − 1) (7.28) × (cid:34)(cid:88) a πi (cid:90) − C V Q − s (cid:48) ζ a ,Q ( s (cid:48) , z )Γ( s − − z )Γ( s − + z ) ζ ∗ (1 + 2 z ) ζ ∗ (1 − z )Γ( − z )Γ( + z ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) dz + V Q − s (cid:48) K + a ,Q ( s (cid:48) ) Γ( s + − s (cid:48) )Γ( s − + s (cid:48) ) (cid:10) V, E ∗ a ( ∗ , − s (cid:48) ) (cid:11) π − s (cid:48) ζ ∗ (3 − s (cid:48) )Γ( s (cid:48) − ) Γ( − s (cid:48) ) (cid:35) . Note the cancellation of ζ ∗ (1 − z ) in the denominator. We see that the above expressionis analytic for s (cid:48) to the right of 1 − C and to the left of 1 + C . If we then take s (cid:48) to beto the left of Re s (cid:48) = 1, we can shift the line of integration back to Re z = 0, passing overthe pole at z = s (cid:48) − S int ( s, s (cid:48) ) = (4 π ) k Γ(1 − s )2Γ( s + k − 1) (7.29) × (cid:34)(cid:88) a πi (cid:90) − C σ V Q − s (cid:48) ζ a ,Q ( s (cid:48) , z )Γ( s − − z )Γ( s − + z ) ζ ∗ (1 + 2 z ) ζ ∗ (1 − z )Γ( − z )Γ( + z ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) dz + Γ( s + − s (cid:48) )Γ( s − + s (cid:48) ) Q s (cid:48) π − s (cid:48) ζ ∗ (3 − s (cid:48) )Γ( s (cid:48) − ) Γ( − s (cid:48) ) × (cid:16) K + a ,Q ( s (cid:48) ) (cid:10) V, E ∗ a ( ∗ , − s (cid:48) ) (cid:11) + K − a ,Q ( s (cid:48) ) (cid:10) V, E ∗ a ( ∗ , s (cid:48) − ) (cid:11)(cid:17)(cid:105) which, still for fixed s at least (cid:15) away from − r for r ≥ 0, is a meromorphic functionin s (cid:48) defined for Re s (cid:48) < 1. We observe that S int only has poles in s (cid:48) due to Γ( s + − s (cid:48) )and Γ( s − + s (cid:48) ), and we will show next that only the latter function contributes polesin S cts . Indeed, after following nearly identical reasoning as above in resolving the casewhere s = − r for r ≥ 0, we derive the following meromorphic expansion for S cts , for all − k > Re s = σ > − A and all s (cid:48) with Re s (cid:48) = σ (cid:48) such that the integral component isconvergent, S cts = (4 π ) k Γ(1 − s )2Γ( s + k − × (cid:88) a πi (cid:90) − C σ,σ (cid:48) V Q − s (cid:48) ζ a ,Q ( s (cid:48) , z )Γ( s − − z )Γ( s − + z ) ζ ∗ (1 + 2 z ) ζ ∗ (1 − z )Γ( − z )Γ( + z ) (cid:104) V, E ∗ a ( ∗ , + z ) (cid:105) dz + Ψ( s, w ) , (7.30)where ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 47 Ψ( s, w ):= (4 π ) k Γ(1 − s )2Γ( s + k − (cid:88) a (cid:20) (cid:98) − σ (cid:99) (cid:88) (cid:96) =0 (cid:32) V Q − s (cid:48) ζ a ,Q (cid:0) s (cid:48) , − + s + (cid:96) (cid:1) ζ ∗ (2 − s − (cid:96) ) ζ ∗ (2 s + 2 (cid:96) ) (cid:104) V, E ∗ a ( ∗ , s + (cid:96) ) (cid:105) + (1 − δδδ σ,(cid:96) ) V Q − s (cid:48) ζ a ,Q (cid:0) s (cid:48) , − s − (cid:96) (cid:1) ζ ∗ (2 s + 2 (cid:96) ) ζ ∗ (2 − s − (cid:96) ) (cid:104) V, E ∗ a ( ∗ , − s − (cid:96) ) (cid:105) (cid:33) ( − (cid:96) Γ(2 s + (cid:96) − (cid:96) !Γ( s + (cid:96) )Γ(1 − s − (cid:96) )+ δδδ (cid:48) σ (cid:48) Γ( s + − s (cid:48) )Γ( s − + s (cid:48) ) Q s (cid:48) π − s (cid:48) ζ ∗ (3 − s (cid:48) )Γ( s (cid:48) − ) Γ( − s (cid:48) ) × (cid:16) K + a ,Q ( s (cid:48) ) (cid:10) V, E ∗ a ( ∗ , − s (cid:48) ) (cid:11) + (1 − δδδ σ (cid:48) , − ) K − a ,Q ( s (cid:48) ) (cid:10) V, E ∗ a ( ∗ , s (cid:48) − ) (cid:11)(cid:17) (cid:21) . (7.31)Here K ± a ,Q ( s (cid:48) ) is the ratio of Dirichlet polynomials defined by (7.6), δδδ σ,(cid:96) is as in (4.13) andsimilarly δδδ (cid:48) x = (cid:40) x > 11 when x ≤ . (7.32)Morally, we define C σ,σ (cid:48) to be (0) or a curve slightly to the right of (0) such that theintegrand of the integral portion of S cts doesn’t run afoul of any poles. More specificallywe define it to be C σ,σ (cid:48) = (0) when σ (cid:48) (cid:54) = 1 and σ (cid:54) = − r for r ∈ Z ≥ C when σ (cid:48) = 1 and σ = − r for r ∈ Z ≥ (cid:15) σ C when σ = 1 and σ (cid:54) = − r for r ∈ Z ≥ (cid:15) (cid:48) σ (cid:48) C when σ (cid:48) (cid:54) = 1 and σ = − r for r ∈ Z ≥ (7.33)where C is as in the description accompanying (4.10). We choose 1 > (cid:15) d , (cid:15) (cid:48) d (cid:48) > | σ − + r | > | Re (cid:15) σ C ( x ) | for all r ∈ Z ≥ and | σ (cid:48) − | > | Re (cid:15) σ (cid:48) C ( x ) | in the respective cases.As remarked before, the right hand side of (7.30) converges in a superset of the regionwhere S cusp ( s, s (cid:48) ) converges. We also note that S cts ( s, s (cid:48) ) indeed has simple poles at s = − r for r ∈ Z ≥ and poles at (cid:37) − r for r ∈ Z ≥ where (cid:37) is a non-trivial zero of ζ . The only other poles to consider are the potential simple poles at s (cid:48) = − s − (cid:96) and s (cid:48) = + s + (cid:96) for (cid:96) ≥ 0. The latter poles should not exist, as we already noted abovethat they cannot appear when Re s (cid:48) > 1. Indeed, suppose that Re s (cid:48) ≤ s (cid:48) = + s + (cid:96) , then Re s ≤ − (cid:96) , and a simple omitted computation shows that thesepoles do indeed cancel. We similarly compute the residues when s (cid:48) = − s − (cid:96) , although we no longer necessarily assume that Re s (cid:48) ≤ s (cid:48) = − s − (cid:96) S cts ( s, s (cid:48) ) = (4 π ) k Γ(1 − s )2Γ( s + k − 1) (7.34) × (cid:88) a (cid:34) δδδ (cid:48) + σ + (cid:96) V ( − (cid:96) Γ(2 s + (cid:96) − Q − ( − s − (cid:96) ) (cid:96) ! π s + (cid:96) − ζ ∗ (2 s + 2 (cid:96) )Γ( s + (cid:96) )Γ(1 − s − (cid:96) ) × (cid:16) K − a ,Q ( − s − (cid:96) ) (cid:104) V, E ∗ a ( ∗ , − s − (cid:96) ) (cid:105) +(1 − δδδ σ,(cid:96) ) K + a ,Q ( − s − (cid:96) ) (cid:104) V, E ∗ a ( ∗ , s + (cid:96) ) (cid:105) (cid:17) + δδδ (cid:48) − σ − (cid:96) V ( − (cid:96) Γ(2 s + (cid:96) − (cid:96) ! Q − s − (cid:96) π s + (cid:96) − ζ ∗ (2 s + 2 (cid:96) )Γ(1 − s − (cid:96) ) Γ( s + (cid:96) ) × (cid:16) K + a ,Q ( − s − (cid:96) ) (cid:104) V, E ∗ a ( ∗ , s + (cid:96) ) (cid:105) +(1 − δδδ (cid:48) + σ + (cid:96) ) K − a ,Q ( − s − (cid:96) ) (cid:104) V, E ∗ a ( ∗ , − s − (cid:96) ) (cid:105) (cid:17)(cid:105) = (4 π ) k Γ(1 − s )2Γ( s + k − (cid:88) a (cid:34) V ( − (cid:96) Γ(2 s + (cid:96) − Q − ( − s − (cid:96) ) (cid:96) ! π s + (cid:96) − ζ ∗ (2 s + 2 (cid:96) )Γ( s + (cid:96) )Γ(1 − s − (cid:96) ) × (cid:16) K + a ,Q ( − s − (cid:96) ) (cid:104) V, E ∗ a ( ∗ , s + (cid:96) ) (cid:105) + K − a Q ( − s − (cid:96) ) (cid:104) V, E ∗ a ( ∗ , − s − (cid:96) ) (cid:105) (cid:17)(cid:105) . The above computation gives us (7.7) of the proposition.7.2. Removing the poles of Z Q ( s, w ) . We have now determined the location of thepoles of S cts ( s, s (cid:48) ), and these poles are cancelled by B A ( s )Γ( s (cid:48) − + s )Γ(2 s − , where B A ( s ) := [ A (cid:99) (cid:89) (cid:96) =0 ζ (2 s + 2 (cid:96) )( s + (cid:96) − ) . It follows that B A ( s )Γ( s (cid:48) − + s )Γ(2 s − S cts ( s, s (cid:48) )is analytic in its region of convergence. This region includes all ( s, w ) with c < Re s < Re w sufficiently large, and all ( s, w ) with c < Re s < − k and Re w > 1, and all( s, w ) with Re s > Re w > D ( s ; h ) given inProposition 5.1 and define ˜ D ( s ; h ) by˜ D ( s ; h ) = D ( s ; h ) e s − (cid:88) t j (cid:54) =0 , ≤ r< − c − Im ( t j ) c r,j h r − it j ρ j ( h ) e ( − r + it j ) s − ( − r + it j ) . (7.35) ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 49 The function e s has been included to insure that the sum over t j above converges. A poly-nomially decaying function in the imaginary part of s would suffice, but e s is convenientas it introduces no new poles.By construction, ˜ D ( s ; h ) is analytic for all s with Re s > c . Now define˜ Z Q ( s, w ) = ( (cid:96) (cid:96) ) ( k − / (cid:88) h ≥ ˜ D ( s ; h Q )( h Q ) w +( k − / . (7.36)By Proposition 5.1, for c < Re s < 2, the sum over h in Z Q ( s, w ) converges when s atleast (cid:15) away from poles and when Re w + ( k − / > − c . The sum (cid:88) h ≥ (cid:88) t j , ≤ r< − c − Im ( t j ) c r,j ( h Q ) r − it j ρ j ( h Q )) e ( − r + it j ) ( h Q ) w +( k − / ( s − + r − it j )also converges in the larger region where Re w + ( k − / > / θ − c . Thus ˜ Z Q ( s, w )converges and ˜ Z Q ( s, w ) is analytic along the lines ( − r + it j , w ) in the region c < Re s < Re w + ( k − / > − c .By (7.35) ˜ Z Q ( s, w ) = Z Q ( s, w ) e s − P Q ( s, w ) , (7.37)where P Q ( s, w ) = ( (cid:96) (cid:96) ) ( k − / (cid:88) t j (cid:54) =0 , ≤ r< − c − Im ( t j ) ,h ≥ c r,j ( h Q ) r − it j ρ j ( h Q ) e ( − r + it j ) ( h Q ) w +( k − / ( s − ( − r + it j )) (7.38)= ( (cid:96) (cid:96) ) ( k − / (cid:88) t j (cid:54) =0 , ≤ r< − c − Im ( t j ) c r,j L Q ( w + ( k − / − r + it j , u j ) e ( − r + it j ) s − ( − r + it j ) . By construction, when t j (cid:54) = 0,Res s = − r + it j Z Q ( s, w ) e s = Res s = − r + it j P Q ( s, w ) (7.39)= ( (cid:96) (cid:96) ) ( k − / c r,j L Q ( w + ( k − / − r + it j , u j ) e ( − r + it j ) . The interchange in the order of summation above is allowed if the real part of w issufficiently large. Once interchanged, it is clear that, given the analytic continuation of L Q ( w + ( k − / − r + it j , u j ), the summation converges for any fixed value of w . Thusthe sum defining P Q ( s, w ) converges and is, again by construction, analytic for all points s, w ∈ C with Re s > c , except for polar lines at ( s, w ) = ( − r + it j , w ). If t j = 0 forsome Maass form of this level then we have double poles at s = − r for r ≥ Z Q ( s, w ) has had its poles of the form ( − r + it j , w ) removed, and multiplyingby B A ( s ) / (Γ( s (cid:48) + s − )Γ(2 s − spectrum. By the previous analysis, it is also convergent in a region including all ( s, w )with c < Re s < Re w sufficiently large, and all ( s, w ) with c < Re s < − k and Re w > 1, and all ( s, w ) with Re s > Re w > 1. The convex hull of this tubedomain is all ( s, w ) with Re w > Re s > c . Consequently, by Bˆochner’s theorem,˜ Z Q ( s, w ) is analytic in this convex hull. This completes the proof of the proposition. (cid:3) Double shifted sums Removing the weighting factor. The unweighted but smoothed sum we are in-terested in estimating is S Q ( x, y ) = (cid:88) (cid:96) m (cid:96) m hQh,m ≥ A ( m ) ¯ B ( m ) G ( m /y ) G ( m /x ) . The above sum is related to Z Q ( s, w ) via a triple inverse Mellin transform, as set forth inthe following proposition. Proposition 8.1. Suppose G ( x ) , G ( x ) are smooth with compact support in the interval [1 , , with g ( s ) , g ( s ) denoting the corresponding Mellin transforms as before. For x (cid:29) and y (cid:29) , S Q ( x, y ) = (cid:18) πi (cid:19) (cid:90) ( γ ) (cid:90) ( γ ) (cid:90) ( γ ) Z Q ( s + w − u, u + (1 − k ) / k − / w ) × Γ(( k − / w − u )Γ( u ) (cid:96) w (cid:96) s x s y w g ( w ) g ( s ) dudwds. (8.1) Here γ , γ , γ are chosen so that Re s + w − u > , Re u + (1 − k ) / > , and Re ( k − / w − u > . In particular we choose γ = ( k + 1) / , γ = 1 + 2 (cid:15) , γ = ( k + 1) / (cid:15) ,for any (cid:15) > .Proof. Recall the identity (6.4). Changing the variables in (7.3), and setting β = ( k − / w , we have, for any γ satisfying 0 < γ < Re β ,12 πi (cid:90) ( γ ) Z Q ( s + w − u, u + (1 − k ) / u )Γ(( k − / w − u ) (cid:96) w (cid:96) s du (8.2)= (cid:88) m ,h ≥ (cid:96) m (cid:96) m h Q A ( m ) ¯ B ( m ) 12 πi (cid:90) ( γ ) Γ( u )Γ( k − + w − u ) (cid:96) w (cid:96) s (cid:16) h Q(cid:96) m (cid:17) ( k − / du ( (cid:96) m ) s + w − u ( h Q ) u = Γ( k − + w ) (cid:88) m ,h ≥ (cid:96) m (cid:96) m h Q A ( m ) ¯ B ( m ) (cid:96) w (cid:96) s ( (cid:96) m ) s + w (cid:18) h Q(cid:96) m (cid:19) − w − ( k − / (cid:18) h Q(cid:96) m (cid:19) ( k − / = Γ( k − + w ) (cid:88) m ,h ≥ (cid:96) m (cid:96) m h Q A ( m ) ¯ B ( m ) (cid:96) w (cid:96) s ( (cid:96) m ) s + w (cid:16) h Q(cid:96) m (cid:17) w = Γ( k − + w ) (cid:88) m ,h ≥ (cid:96) m (cid:96) m h Q A ( m ) ¯ B ( m ) m s m w . ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 51 The proposition then follows, after setting γ = γ and applying the inverse Mellin trans-form (cid:18) πi (cid:19) (cid:90) (cid:90) ( γ )( γ ) g ( w ) g ( s ) x s y w dwds to both sides. (cid:3) Having related S Q ( x, y ) to Z Q ( s + w − u, u + (1 − k ) / 2) we will now derive an estimate for S Q ( x, y ) from the analytic properties of Z Q ( s + w − u, u +(1 − k ) / 2) and prove Theorem 1.3from the introduction. Proof of Theorem 1.3. Changing variables in the preceding Proposition, S Q ( x, y ) = (cid:18) πi (cid:19) (cid:90) (cid:90) (cid:90) (1+ (cid:15) )(1+2 (cid:15) )(1+ (cid:15) ) Z Q ( s, u )Γ( w − u )Γ( u + ( k − / w + ( k − / × ( (cid:96) x ) s + u +( k − / − w ( (cid:96) y ) w g ( w ) g ( s + u + ( k − / − w ) dudwds. (8.3)Referring to (7.8), rewrite S cusp ( s, s (cid:48) ) as S cusp ( s, s (cid:48) ) = S cusp , gen ( s, s (cid:48) ) + S cusp,except ( s, s (cid:48) ) , where S cusp,gen ( s, s (cid:48) ) := (4 π ) k Γ(1 − s )2Γ( s + k − (cid:88) Re t j (cid:54) =0 L Q ( s (cid:48) , u j )Γ( s − − it j )Γ( s − + it j ) (cid:104) V, u j (cid:105) Γ( − it j )Γ( + it j ) , and S cusp,except ( s, s (cid:48) ) := (4 π ) k Γ(1 − s )2Γ( s + k − (cid:88) Re t j =0 L Q ( s (cid:48) , u j )Γ( s − − it j )Γ( s − + it j ) (cid:104) V, u j (cid:105) Γ( − it j )Γ( + it j ) . Referring to Proposition 7.1, we define Z Q, gen ( s, u ) by Z Q ( s, u ) = Z Q, gen ( s, u ) + Z Q, except ( s, u ) , where Z Q, except ( s, u ) := ( (cid:96) (cid:96) ) ( k − / S cusp,except ( s, s (cid:48) ) . With this notation, we rewrite (8.3) as S Q ( x, y ) = S Q, gen ( x, y ) + S Q, except ( x, y ) , where S Q, except ( x, y ) = (cid:18) πi (cid:19) (cid:90) (cid:90) (cid:90) (1+ (cid:15) )(1+2 (cid:15) )(1+ (cid:15) ) Z Q, except ( s, u )Γ( w − u )Γ( u + ( k − / w + ( k − / × ( (cid:96) x ) s + u +( k − / − w ( (cid:96) y ) w g ( w ) g ( s + u + ( k − / − w ) dudwds (8.4) and S Q, gen ( x, y ) = (cid:18) πi (cid:19) (cid:90) (cid:90) (cid:90) (1+ (cid:15) )(1+2 (cid:15) )(1+ (cid:15) ) Z Q, gen ( s, u )Γ( w − u )Γ( u + ( k − / w + ( k − / × ( (cid:96) x ) s + u +( k − / − w ( (cid:96) y ) w g ( w ) g ( s + u + ( k − / − w ) dudwds. (8.5)To estimate S Q, except ( x, y ) we first move the u line of integration to Re u = (cid:15) − ( k − / Re s (cid:48) = 1 / (cid:15) . Moving the w line to Re w = 0, and referring to (7.18), the definition of Z Q, except ( s, u ), and (8.4) we find that S Q, except ( x, y ) (cid:28) Q θ − / − (cid:15) L (cid:15) x (cid:15) . (8.6)To estimate S Q, gen ( x, y ), we refer to (8.5) and move the s line of integration to Re s = − k − (cid:15) . Applying Proposition 7.1 and using the above notation we write S Q, gen ( x, y ) = S (1) Q ( x, y ) + S (2) Q ( x, y ) + S (3) Q ( x, y ) , (8.7)where S (1) Q ( x, y ) := (cid:18) πi (cid:19) (cid:90) (cid:90) (cid:90) ( − k − (cid:15) )(1+2 (cid:15) )(1+ (cid:15) ) ( Z Q, gen ( s, u ) − Ψ( s, u ))Γ( w − u )Γ( u + ( k − / w + ( k − / × ( (cid:96) x ) s + u +( k − / − w ( (cid:96) y ) w g ( w ) g ( s + u + ( k − / − w ) dudwds (8.8) S (2) Q ( x, y ) := (cid:18) πi (cid:19) (cid:88) Re t j (cid:54) =00 ≤ r ≤ k (cid:90) (cid:90) (1+2 (cid:15) )(1+ (cid:15) ) c r,j L Q (cid:0) − r + it j + u + k − , u j (cid:1) Γ( w − u )Γ( u + k − )Γ( w + k − ) × ( (cid:96) x ) − r + it j + u + k − w ( (cid:96) y ) w g ( w ) g ( − r + it j + u + k − w ) dudw (8.9)and, after a slight shift of the w integral, S (3) Q ( x, y ) = (cid:18) πi (cid:19) (cid:90) (cid:90) (cid:90) (1+ (cid:15) )(1+3 (cid:15) )(1+ (cid:15) ) Ψ( s, u )Γ( w − u )Γ( u + ( k − / w + ( k − / × ( (cid:96) x ) s + u +( k − / − w ( (cid:96) y ) w g ( w ) g ( s + u + ( k − / − w ) dudwds. (8.10)The last integral, S (3) Q ( x, y ) is really just a shorthand for the residual contribution dueto the continuous part of the spectrum, since Ψ( s, u ) is defined piecewise. To analyze S (1) Q ( x, y ), let s (cid:48) = s + u + k − 1, so Re s (cid:48) = . By (7.10) in Proposition 7.1 we have Z Q, gen ( s, u ) − Ψ( s, u ) (cid:28) Q θ − Re s (cid:48) + (cid:15) (1 + | s | ) A (cid:48)(cid:48) (1 + | s (cid:48) | ) (cid:15) . As this has polynomial growth in Im s and Im u , and there is decay faster than anypolynomial due to the factors g ( w ) g ( s + u + ( k − / − w ), it follows that the double ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 53 integral over s and u in S (1) Q ( x, y ) converges and is bounded above by Q θ − + (cid:15) . The integralover w also converges, and its size is governed by the factor( (cid:96) x ) s + u +( k − / − w ( (cid:96) y ) w = ( (cid:96) x ) s + u +( k − / (( (cid:96) y ) / ( (cid:96) x )) w . (8.11)As y (cid:29) x and (cid:96) ∼ (cid:96) ∼ L , it is advantageous to move w down past Re w = Re u = 1 + (cid:15) ,picking up a pole at w = u . Thus S (1) Q ( x, y ) is finally bounded above by Q θ − + (cid:15) L (cid:15) y ( y/x ) (cid:15) . (8.12)This is weaker than (8.6) above, unless x = y . Remark 8.2. The upper bound (8.12), coming from the pole at w = u is the bottleneckin this proof. If would be desirable to move the u line of integration as far past 1 + (cid:15) as possible. The difficulty is that this can only be done if one possesses an upper boundfor Z Q, gen ( s, u ) − Ψ( s, u ) in the region Re u < 1. This is possible, using the meromorphiccontinuation of Z Q, gen ( s, u ) given in [25].For S (2) Q ( x, y ), move the u line of integration to Re u = 1 + r − k − Re it j , so Re ( − r + it j + u + k ) = 1. Note that for r ≥ 0, the argument of that gamma function is Re u + ( k − / Re r − k − Re it j + ( k − / ≥ , so the first pole of Γ( u +( k − / 2) is not passed. The double integral over u, w in S (2) Q ( x, y )converges, as does the sum over the t j . Referring to (7.18), we see that an upper boundfor S (2) Q ( x, y ) is given by Q θ − + (cid:15) L (cid:15) y ( y/x ) r − k − Re it j , which is an improvement over (8.12) as r − k − Re it j ≤ S (3) Q . Changing variables from u to s (cid:48) , with s (cid:48) = s + u + k − 1, gives us S (3) Q ( x, y ) = (cid:18) πi (cid:19) (cid:90) (cid:90) (cid:90) (1+ (cid:15) )(1+3 (cid:15) )(1+ k +2 (cid:15) ) Ψ( s, s (cid:48) − s − k + 1)Γ( w − s (cid:48) + s + k − s (cid:48) − s + )Γ( w + ( k − / × ( (cid:96) x ) s (cid:48) − w + ( (cid:96) y ) w g ( w ) g ( s (cid:48) − w + ) ds (cid:48) dwds. We now move the s (cid:48) line of integration to Re s (cid:48) = + 2 (cid:15) . Here Re s (cid:48) − s + = + 2 (cid:15) − − (cid:15) + = (cid:15) , so the pole of Γ( s (cid:48) − s + ) where s (cid:48) − s + = 0 is not passed over. However,from (7.31) we do get that S (3) Q ( x, y ) = (cid:18) πi (cid:19) (cid:90) (cid:90) (cid:90) (1+ (cid:15) )(1+3 (cid:15) )( +2 (cid:15) ) Ψ( s, s (cid:48) − s − k + 1)Γ( w − s (cid:48) + s + k − s (cid:48) − s + )Γ( w + ( k − / × ( (cid:96) x ) s (cid:48) − w + ( (cid:96) y ) w g ( w ) g ( s (cid:48) − w + ) ds (cid:48) dwds + R ( x, y ) , (8.13) where R ( x, y ) := (cid:88) a (cid:18) πi (cid:19) (cid:90) (cid:90) (cid:90) (1+ (cid:15) )(1+3 (cid:15) )(1) (4 π ) k Γ(1 − s )Γ( s + − s (cid:48) )Γ( s − + s (cid:48) )2 Q s (cid:48) Γ( s + k − π − s (cid:48) ζ ∗ (3 − s (cid:48) )Γ( s (cid:48) − ) Γ( − s (cid:48) ) × (cid:16) K + a ,Q ( s (cid:48) ) (cid:10) V, E ∗ a ( ∗ , − s (cid:48) ) (cid:11) + K − a ,Q ( s (cid:48) ) (cid:10) V, E ∗ a ( ∗ , s (cid:48) − ) (cid:11)(cid:17) × Γ( w − s (cid:48) + s + k − s (cid:48) − s + )Γ( w + ( k − / 2) ( (cid:96) x ) s (cid:48) − w + ( (cid:96) y ) w g ( w ) g ( s (cid:48) − w + ) ds (cid:48) dwds and K ± a ,Q ( s (cid:48) ) is given by (7.5) and (7.6). By changing variables s (cid:48) → z and the orderof integration we get R ( x, y ) = (cid:88) a (cid:18) πi (cid:19) (cid:90) (cid:90) (cid:90) (0)(1+ (cid:15) )(1+3 (cid:15) ) (4 π ) k π + z Γ(1 − s )Γ( s − − z )Γ( s − + z )2 Q z Γ( s + k − ζ ∗ (1 − z )Γ( + z ) Γ( − z ) × (cid:16) K + a ,Q (1 + z ) (cid:10) V, E ∗ a ( ∗ , − z ) (cid:11) + K − a ,Q (1 + z ) (cid:10) V, E ∗ a ( ∗ , + z ) (cid:11)(cid:17) × Γ( w − k − z + s )Γ(3 / z − s )Γ( w + ( k − / 2) ( (cid:96) x ) / z − w ( (cid:96) y ) w g ( w ) g (3 / z − w ) dwdsdz. Now taking the first integral component of S (3) Q ( x, y ) as given in (8.13), we collect thedifferent components of Ψ( s, s (cid:48) − s − k + 1) to get that S (3) Q ( x, y ) = (cid:88) a , ≤ r ≤ k (cid:18) πi (cid:19) (cid:90) (cid:90) (cid:90) ( − r )(1+3 (cid:15) )( +2 (cid:15) ) ( − r (4 π ) k V Q − s (cid:48) Γ(1 − s )Γ(2 s + r − r !Γ( s + k − s + r )Γ(1 − s − r ) × (cid:32) ζ a ,Q (cid:0) s (cid:48) , − + s + r (cid:1) (cid:104) V, E ∗ a ( ∗ , s + r ) (cid:105) ζ ∗ (2 − s − r ) ζ ∗ (2 s + 2 r ) + ζ a ,Q (cid:0) s (cid:48) , − s − r (cid:1) (cid:104) V, E ∗ a ( ∗ , − s − r ) (cid:105) ζ ∗ (2 s + 2 r ) ζ ∗ (2 − s − r ) (cid:33) × Γ( w − s (cid:48) + s + k − s (cid:48) − s + )Γ( w + ( k − / 2) ( (cid:96) x ) s (cid:48) − w + ( (cid:96) y ) w g ( w ) g (3 / z − w )d s (cid:48) d w d s + R ( x, y ) . ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 55 By substituting s → − r + z and shifting the line of integration for s (cid:48) to Re s (cid:48) = weget S (3) Q ( x, y ) = (cid:88) a , ≤ r ≤ k (cid:18) πi (cid:19) (cid:90) (cid:90) (cid:90) (0)(1+3 (cid:15) )( ) ( − r (4 π ) k V Q − s (cid:48) Γ( − z )Γ(2 z − r )2 r !Γ( z + k − r − )Γ( + z )Γ( − z ) × (cid:32) ζ a ,Q ( s (cid:48) , z ) (cid:104) V, E ∗ a (cid:0) ∗ , + z (cid:1) (cid:105) ζ ∗ (1 − z ) ζ ∗ (1 + 2 z ) + ζ a ,Q ( s (cid:48) , − z ) (cid:104) V, E ∗ a ( ∗ , − z ) (cid:105) ζ ∗ (1 + 2 z ) ζ ∗ (1 − z ) (cid:33) × Γ( w + z − s (cid:48) − r + k − )Γ( s (cid:48) − r − z )Γ( w + ( k − / 2) ( (cid:96) x ) s (cid:48) − w + ( (cid:96) y ) w g ( w ) g ( s (cid:48) − w + ) ds (cid:48) dwdz + R ( x, y ) . Our method for bounding each of the integrals above, except for R ( x, y ), is completelyanalogous to how we bounded S (2) Q ( x, y ). By shifting the line of integration w left pastthe pole at w + z − s (cid:48) − r + k − = 0, we see that our bound for these integrals is also animprovement on (8.12).The z, s and w integrals in R ( x, y ), as in (8.13), are all convergent. By shifting the lineof integration for z to Re z = − − (cid:15) we get a bound that is an improvement on (8.12)and our proof is complete. (cid:3) Application to subconvexity Our amplification argument is modeled on that of Blomer in [5], which is in turn basedon that of [14]. Recall we have fixed a positive integer Q , with ( Q, N ) = 1. Recall alsothat we have fixed N , (cid:96) , (cid:96) ∈ N , with N , (cid:96) , (cid:96) square free, ( N , (cid:96) (cid:96) ) = 1 and have set N = N (cid:96) (cid:96) / ( (cid:96) , (cid:96) ). In this section we will add the requirement that (cid:96) , (cid:96) be prime. Let χ be a primitive character mod Q then for Re s > L ( s, f ⊗ χ ) = (cid:88) m ≥ A ( m ) χ ( m ) m s . Using the approximate functional equation, we have that L ( , f ⊗ χ ) = (cid:88) m A ( m ) χ ( m ) √ m V (cid:18) mQ √ N (cid:19) + ω (cid:88) m ¯ A ( m ) ¯ χ ( m ) √ m V (cid:18) mQ √ N (cid:19) , where ω is the root number. Let B χ ( x ) = (cid:88) m A ( m ) χ ( m ) G (cid:16) mx (cid:17) , where G is smooth of compact support in [1 , 2] and x is a parameter. By a smooth dyadicpartition of unity, and summation by parts, we have L ( , f ⊗ χ ) (cid:28) max x (cid:28) Q (cid:15) Q − B χ ( x ) . (9.1)Set L = Q / − θ/ (cid:15) , and for any x (cid:28) Q , consider the amplified sum S = (cid:88) ψ ( Q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) m A ( m ) ψ ( m ) G (cid:16) mx (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) (cid:96) ∼ L,(cid:96) prime , ( (cid:96),N )=1 ¯ χ ( (cid:96) ) ψ ( (cid:96) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (9.2)Notice that the term with ψ = χ contributes | B χ ( x ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) (cid:96) ∼ L,(cid:96) prime , ( (cid:96),N )=1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The number of (cid:96) ∼ L with (cid:96) prime is asymptotic to L/ log( L ) ∼ Q / − θ/ (cid:15) log( Q / − θ/ (cid:15) ) , so taking only the contribution to S from ψ = χ gives B χ ( x ) (cid:28) S Q − / θ/ . (9.3)Combining this with (9.1) we now have L ( , f ⊗ χ ) (cid:28) max x (cid:28) Q Q − / θ/ S . (9.4)If we open the squares in S , and bring the sum over ψ inside, we obtain, S = (cid:88) m (cid:88) m (cid:88) (cid:96) (cid:88) (cid:96) A ( m ) A ( m ) G (cid:16) m x (cid:17) G (cid:16) m x (cid:17) χ ( (cid:96) ) χ ( (cid:96) ) (cid:88) ψ ψ ( (cid:96) m ) ψ ( (cid:96) m )= φ ( Q ) (cid:88) m ,m ,(cid:96) ,(cid:96) , ( (cid:96) m ,Q )=1 (cid:96) m ≡ (cid:96) m (mod Q ) A ( m ) A ( m ) G (cid:16) m x (cid:17) G (cid:16) m x (cid:17) χ ( (cid:96) ) χ ( (cid:96) ) . (9.5) ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 57 We can remove the restriction ( (cid:96) m , Q ) = 1 by letting S = φ ( Q ) (cid:88) a mod Q ( a,Q )=1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) m (cid:88) (cid:96) ∼ L,(cid:96) prime , ( (cid:96),N )=1 m(cid:96) ≡ a mod Q A ( m ) G (cid:16) mx (cid:17) χ ( (cid:96) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (9.6) ≤ φ ( Q ) (cid:88) a mod Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) m (cid:88) (cid:96) ∼ L,(cid:96) prime , ( (cid:96),N )=1 m(cid:96) ≡ a mod Q A ( m ) G (cid:16) mx (cid:17) χ ( (cid:96) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = φ ( Q ) (cid:88) a (cid:88) m (cid:88) m (cid:88) (cid:96) (cid:88) (cid:96) m (cid:96) ≡ m (cid:96) ≡ a mod Q A ( m ) A ( m ) G (cid:16) m x (cid:17) G (cid:16) m x (cid:17) χ ( (cid:96) ) χ ( (cid:96) ) , which can be expressed as, S ≤ φ ( Q ) (cid:88) (cid:96) ,(cid:96) ∼ L primes( (cid:96) (cid:96) ,N )=1 χ ( (cid:96) ) χ ( (cid:96) ) (cid:88) m ,m m (cid:96) ≡ m (cid:96) mod Q A ( m ) A ( m ) G (cid:16) m x (cid:17) G (cid:16) m x (cid:17) . This sum, over all m , m such that (cid:96) m ≡ (cid:96) m (mod Q ), can be broken into threepieces, two of which are treatable by our Proposition 8.1. S (cid:28) φ ( Q ) (cid:88) (cid:96) (cid:88) (cid:96) χ ( (cid:96) ) χ ( (cid:96) )( S + S + S ) , (9.7)where S = (cid:88) m ≥ (cid:96) m = (cid:96) m A ( m ) ¯ A ( m ) G (cid:16) m x (cid:17) G (cid:16) m x (cid:17) ,S = (cid:88) m ,h ≥ (cid:96) m = (cid:96) m + h Q A ( m ) ¯ A ( m ) G (cid:16) m x (cid:17) G (cid:16) m x (cid:17) , and S = (cid:88) m ,h ≥ (cid:96) m = (cid:96) m + h Q A ( m ) ¯ A ( m ) G (cid:16) m x (cid:17) G (cid:16) m x (cid:17) . It is easily verified that φ ( Q ) (cid:88) (cid:96) (cid:88) (cid:96) ¯ χ ( (cid:96) ) χ ( (cid:96) ) S (cid:28) LQ (cid:15) . The sums S , S are bounded using Theorem 1.3, with x = y (cid:28) Q (cid:15) . This gives us S , S (cid:28) Q θ − x (cid:15) L (cid:15) (cid:28) Q + θ + (cid:15) L (cid:15) . Substituting, we have for their contribution: φ ( Q ) (cid:88) (cid:96) (cid:88) (cid:96) ¯ χ ( (cid:96) ) χ ( (cid:96) )( S + S ) (cid:28) QL Q + θ + (cid:15) L (cid:15) (cid:28) Q / θ + (cid:15) L (cid:15) . Referring to (9.7) we have, S (cid:28) φ ( Q ) (cid:88) (cid:96) (cid:88) (cid:96) ¯ χ ( (cid:96) ) χ ( (cid:96) )( S + S + S ) (cid:28) Q (cid:15) L + Q / θ + (cid:15) L (cid:15) (cid:28) Q / − θ/ (cid:15) . Finally, referring to (9.4), L ( , f ⊗ χ ) (cid:28) max x (cid:28) Q (cid:15) Q − / θ/ S (cid:28) Q / θ/ (cid:15) . This completes the proof of Theorem 1.1. Appendix: Estimates of triple products of automorphic functions II Andre Reznikov Abstract. We prove a sharp bound for the average value of the triple product ofmodular functions for the Hecke subgroup Γ ( N ). Our result is an extension of the mainresult in [4] to a fixed cuspidal representation of the adele group P GL ( A ). Appendix Maass forms. We recall the setup of [4] which should be read in conjunctionwith this appendix. Let Y be a compact Riemann surface with a Riemannian metricof constant curvature − dv . The correspondingLaplace-Beltrami operator is non-negative and has purely discrete spectrum on the space L ( Y, dv ) of functions on Y . We will denote by 0 = µ < µ ≤ µ ≤ ... its eigenvaluesand by φ i the corresponding eigenfunctions (normalized to have L norm one). In thetheory of automorphic forms the functions φ i are called automorphic functions or Maassforms (after H. Maass, [38]). We write µ i = (1 − λ i ) / φ i = φ λ i as is customary inrepresentation theory of the group P GL ( R ).For any three Maass forms φ i , φ j , φ k we define the following triple product or tripleperiod: c ijk = (cid:90) Y φ i φ j φ k dv . (10.1) Partially supported by the ERC grant 291612, by the ISF grant 533/14, and by the Minerva Centerat ENI ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 59 One would like to bound the coefficient c ijk as a function of eigenvalues µ i , µ j , µ k . Inparticular, we would like to find bounds for these coefficients when one or more of theseindices tend to infinity. The study of these triple coefficients goes back to pioneering worksof Rankin and Selberg (see [42], [47]), and reappeared in celebrated works of Waldspurger[48] and Jacquet [30] (also see [23], [48], [28]). Recently, an interest in analytic questionsrelated to triple products was initiated in the groundbreaking paper of Sarnak [44] (seealso [19] for the first general result on exponential decay). This was motivated by thewidespread use of triple products in applications (e.g., [45]).In [4] we considered the following particular case of the triple period problem. Namely,we fix two Maass forms φ = φ τ , φ (cid:48) = φ τ (cid:48) as above and consider coefficients defined by thetriple period: c i = (cid:90) Y φφ (cid:48) φ i dv (10.2)as { φ i = φ λ i } run over the orthonormal basis of Maass forms. We note first that one hasexponential decay for the coefficients c i in the parameter | λ i | as i goes to ∞ . For thatreason, one renormalizes coefficients | c i | by an appropriate ratio of Gamma functionsdictated by the Watson formula [48] (see also Appendix in [4] where these factors werecomputed using another point of view). Taking into account the asymptotic behavior ofthese factors, we introduced normalized coefficients | a i | = | λ i | exp (cid:16) π | λ i | (cid:17) · | c i | . (10.3)Under such a normalization, we showed that (cid:88) | λ i |≤ T | a i | ≤ A · T (10.4)for some explicit constant A = A (Γ , φ, φ (cid:48) ). According to the Weyl law, there are ap-proximately cT terms in the above sum, and hence the bound (10.4) is consistent withthe Lindel¨of bound on average (in fact it is not difficult to show that the bound (10.4)essentially is sharp, see [43]).There are various natural questions concerning the bound (10.4) which were not discussedin [4]. These are mostly related to the dependence of the constant A on various parameters(i.e., dependence on the subgroup Γ and forms φ , φ (cid:48) ), and also to the fact that we restrictedthe discussion to Maass forms, leaving aside the case of holomorphic forms. Anotherrestriction of the treatment we presented was the fact that we used in an essential waythe compactness of Y (although the non-compact case was treated by us differently in [2]).All these issues turn out to be important in applications. In this appendix, we answersome of these questions for Hecke congruence subgroups. The methods we employ areelaborations on the method of [2].10.1.1. Hecke subgroups. For an integer N ≥ 1, we consider the Hecke subgroup Γ ( N ) ⊂ P SL ( Z ) of the modular group. We normalize the scalar product on the quotient Riemann surface Y N = Γ ( N ) \ H by (cid:104) f, g (cid:105) Y N = vol H ( Y N ) − (cid:82) Y N f ( z )¯ g ( z ) dµ H where dµ H is thestandard volume element on the upper half plane H (i.e., we normalize the volume element dv Y N on Y N to have the total volume 1). Let φ be a (primitive) Hecke-Maass form forthe group Γ ( N ) for some fixed level N ≥ 1. We assume that φ is normalized by the L -norm || φ || L ( Y N ) = 1. For an integer (cid:96) > 1, denote by (cid:96) φ ( z ) = φ ( (cid:96)z ) the corresponding old form for the Hecke subgroup Γ ( (cid:96)N ). The corresponding function (cid:96) φ also turns outto be L -normalized on Y (cid:96)N with respect to the normalization of measures we choose.This follows easily from the Rankin-Selberg method. For two such Maass forms φ and φ (cid:48) ,we define triple products by c i ( (cid:96) ) = (cid:90) Y (cid:96)N (cid:96) φ (cid:96) φ (cid:48) φ i dv Y (cid:96)N (10.5)as { φ i = φ λ i } run over the orthonormal basis of Maass forms on Y (cid:96)N . We have thecorresponding normalized coefficients | a i ( (cid:96) ) | = | λ i | exp (cid:16) π | λ i | (cid:17) | c i ( (cid:96) ) | . (10.6) Theorem A. There exists an effectively computable constant A such that the followingbound holds for all T ≥ , (cid:88) | λ i |≤ T | a i ( (cid:96) ) | ≤ A · T , (10.7) where the summation is over an orthonormal basis of Maass forms for the group Γ ( (cid:96)N ) .The constant A depends on N , φ and φ (cid:48) , but not on (cid:96) . This could be viewed as a Lindel¨of on the average type bound in two parameters (cid:96) and T .Namely, there are about [Γ ( (cid:96)N ) : Γ(1)] · T terms in the sum as predicted by the Weyl-Selberg law for the surface Y (cid:96)N (assuming the uniformity of the corresponding remainderterm). Hence the resulting bound (10.7) is consistent with the expected Lindel¨of typebound | a i ( (cid:96) ) | (cid:28) ( (cid:96) | λ i | ) (cid:15) under our normalization of measures vol( Y (cid:96)N ) = 1.There is also an analogous contribution from the Eisenstein series. One defines normalizedtriple coefficients a s,κ ( (cid:96) ) arising from triple products (cid:104) (cid:96) φ (cid:96) φ (cid:48) , E κ ( s ) (cid:105) Y (cid:96)N where E κ ( s ) is theEisenstein series associated to a cusp κ of Γ ( (cid:96)N ). We prove the following bound for thefull spectral expansion (cid:88) | λ i |≤ T | a i ( (cid:96) ) | + 12 (cid:88) { κ } (cid:90) T − T | a it,κ ( (cid:96) ) | dt ≤ A · T . (10.8)10.2. The method. The proof we presented in [4] was based on the uniqueness of thetriple product in representation theory of the group P GL ( R ). We review quickly thegeneral ideas behind our proof. It is based on ideas from representation theory (see theseminal book [15], and also [9], [35]). Namely, we use the fact that every automorphicform φ generates an automorphic representation of the group G = P GL ( R ); this means ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 61 that starting from φ we produce a smooth irreducible representation of the group G ina space V and its realization ν : V → C ∞ ( X ) in the space of smooth functions on theautomorphic space X = Γ \ G endowed with the invariant measure of the total mass one.We denote by φ v ( x ) = ν ( v )( x ) the corresponding to v ∈ V automorphic function. TheMaass form corresponds to a unit K -invariant vector e ∈ V , where K = P SO (2) is amaximal compact connected subgroup of G . In our case, we have the family of spaces X (cid:96) = Γ ( (cid:96)N ) \ G (we denote by X = Γ ( N ) \ G ), and the corresponding family ofisometries ν (cid:96) : V → C ∞ ( X (cid:96) ) of the same abstract representation of G . We note thatforms φ ( (cid:96)z ) belong to the same adelic representation of P GL ( A ) generated the newHecke-Maass form φ .The triple product c i = (cid:82) Y φφ (cid:48) φ i dv extends to a G -equivariant trilinear form on thecorresponding automorphic representations l aut : V ⊗ V (cid:48) ⊗ V i → C , where V = V τ , V (cid:48) = V τ (cid:48) , V i = V λ i .Then we use a general result from representation theory that such a G -equivariant trilinearform is unique up to a scalar, i.e., that dim Hom G ( V ⊗ V (cid:48) ⊗ V i , C ) ≤ p -adic GL (2)). This implies that the automorphic form l aut is proportional to anexplicit “model” form l mod which we describe using explicit realizations of representationsof the group G ; it is important that this last form carries no arithmetic information.Thus we can write l aut = a i · l modi for some constant a i and hence c i = l auti ( e τ ⊗ e τ (cid:48) ⊗ e λ i ) = a i · l modi ( e τ ⊗ e τ (cid:48) ⊗ e λ i ), where e τ , e τ (cid:48) , e λ i are K -invariant unit vectors in the automorphicrepresentations V, V (cid:48) , V i corresponding to the automorphic forms φ , φ (cid:48) and φ i .It turns out that the proportionality coefficient a i in the last formula carries important“automorphic” information while the second factor carries no arithmetic information andcan be computed in terms of Euler Γ-functions using explicit realizations of representations V τ , V τ (cid:48) and V λ i (see Appendix to [4]). This second factor is responsible for the exponentialdecay, while the first factor a i has a polynomial behavior in the parameter λ i .In order to bound the quantities a i , we use the fact that they appear as coefficients in thespectral decomposition of the diagonal Hermitian form H ∆ given by H ∆ ( v ⊗ w ) = (cid:90) X | φ v ( x ) φ w ( x ) | dx on the space E = V τ ⊗ V τ (cid:48) . This gives an inequality (cid:80) | a i | H i ≤ H ∆ where H i is aHermitian form on E induced by the model trilinear form l modi : V ⊗ V (cid:48) ⊗ V i → C asabove.Using the geometric properties of the diagonal form and simple explicit estimates of forms H i , we establish the mean-value bound for the coefficients | a i | . Here is where one obtainsthe dependence of the constant A in (10.4) on parameters involved. In the method of [4],we used L theory by averaging the form H ∆ and comparing the resulting form with the L -form. The coefficient A that one obtains in such an argument depends in particular onthe injectivity radius of X . While in certain cases it gives an optimal result, it obviously has two drawbacks. One is related to the possible non-compactness of X since in thecusp the injectivity radius tends to zero. Another problem arises when one considers asequence of subgroups with co-volume going to infinity along the sequence. In that case,the bound which the method of [4] provides for the constant (e.g., A (Γ ( p )) ≤ vol( Y p ) ≈ p ,see [33]) is too weak for many applications. Both of these problems arise in the classicalsetup of Hecke subgroups Γ ( N ). Here we obtain an optimal bound for old forms . We donot know how to obtain similar results for new forms. We will discuss improvements overa trivial bound for new forms elsewhere. Theorem A (see Section 10.1.1) could be viewedas the exact analog of the result in [4] for a fixed adelic representation.10.3. Proof of Theorem A. We have a family of objects ( X (cid:96) , ν (cid:96) , H (cid:96) ∆ , a i ( (cid:96) )) parameter-ized by the level (cid:96) . However the model Hermitian form H i is the same since the abstractrepresentation V of P GL ( R ) does not change. The proof of the bound (10.4) given in[4] was based on the spectral decomposition (cid:80) | a i ( (cid:96) ) | H i ≤ H (cid:96) ∆ of the diagonal form andon the construction of the test vector u = u T ∈ E such that H (cid:96) ∆ ( u ) ≤ aT and H i ( u ) ≥ | λ i | ≤ T . We now construct the test vector independently of (cid:96) . The dependence on (cid:96) is hidden in the automorphic realization ν (cid:96) ( u ) as a function on X (cid:96) .10.3.1. Construction of vector u . We slightly change the construction of the test vector u T given in Section 5.3.2 of [4]. Let us identify the space E = V ⊗ V (cid:48) with a subspaceof smooth functions C ∞ ( R × R ). Choose a smooth non-negative function α ∈ C ∞ ( R )with the support supp( α ) ⊂ [ − ι, ι ] for ι > (cid:82) R α ( x ) dx = 1. Let || α || L ( R ) = c for some c > 0. Consider the diagonal element a T = diag( T − , T ) ∈ G .We define vectors v T = T + τ · π τ ( a T ) α and v (cid:48) T = T + τ (cid:48) · π τ (cid:48) (( ) a T ) α . (10.9)We set our test vector to u T ( x, y ) = v T ( x ) ⊗ v (cid:48) T ( y ). Recall that the action is given by π τ (diag( a − , a ) v ( x ) = | a | − τ v ( a x ), and π τ (( )) v ( x ) = v ( x − || u T || E = c T . We note that geometrically the vector u T is a small non-negativebump function around the point (0 , ∈ R , with the support in the box of the size ιT − , and satisfies (cid:82) R u T ( x, y ) dxdy = 1. A computation identical to the one performedin Section 5.3.4 of [4] gives then for a small enough (but fixed) ι > H i ( u T ) ≥ β withsome explicit β > | λ i | ≤ T (in fact for | τ | , | τ (cid:48) | ≤ T , it does not dependon these either). We remark that the only difference with the construction of the testvector given in [4] is that here we constructed u T with the help of the action of G on V (while in [4] we constructed essentially the same vector explicitly in the model). This willplay a crucial role in our estimate of the corresponding automorphic function.We now need to estimate H (cid:96) ∆ ( u T ). We claim that H (cid:96) ∆ ( u T ) ≤ BT for some explicitconstant B independent of (cid:96) . Since we have H (cid:96) ∆ ( u T ) = (cid:90) X (cid:96) | (cid:96) φ v T (cid:96) φ v (cid:48) T | dx ≤ || (cid:96) φ v T || L ( X (cid:96) ) + 12 || (cid:96) φ v (cid:48) T || L ( X (cid:96) ) , ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 63 it is enough to show that || (cid:96) φ v T || L ( X (cid:96) ) ≤ β (cid:48) T . This would finish the proof of Theorem A(Section 10.1.1) following the argument in Section 4.7 [4]. Since (cid:104) (cid:96) φ v T , (cid:105) X (cid:96) = || (cid:96) φ v T || = cT , it is easy to see that such a bound is sharp. We claim thatsup x ∈ X (cid:96) | (cid:96) φ v T ( x ) | ≤ β (cid:48)(cid:48) T , (10.10)for some β (cid:48)(cid:48) independent of (cid:96) . Here (cid:96) φ v T ( x ) = ν (cid:96) ( v T )( x ). Note that the bound providedby the Sobolev theorem [3] would give a much weaker in (cid:96) bound of the order of ( (cid:96)T ) (see [33] for the corresponding discussion).10.3.2. Supremum norm. Recall that we started with an L -normalized Hecke-Maass form φ on Γ ( N ), and the corresponding isometry of ν : V → C ∞ ( X ) of the principal seriesrepresentation V (cid:39) V τ . We then constructed another isometry ν (cid:96) : V → C ∞ ( X (cid:96) ) by usingthe map (cid:96) φ v ( x ) = ν (cid:96) ( v )( x ) = ν ( v ) (cid:16)(cid:16) (cid:96) (cid:96) − (cid:17) x (cid:17) = φ v (cid:16)(cid:16) (cid:96) (cid:96) − (cid:17) x (cid:17) , (10.11)for any v ∈ V . This relation might be viewed as the relation between functions on G invariant on the left for an appropriate Γ (e.g., for Γ ( N ) and for Γ ( (cid:96)N )). In particular,we see that the supremum of the function (cid:96) φ v on X (cid:96) and that of the function φ v on X areequal for the same vector v ∈ V . Hence it is enough to show that sup x ∈ X | φ v T ( x ) | ≤ β (cid:48)(cid:48) T .In fact, this is obvious since v T = T + τ π ( a T ) α is given by the (scaled) action of G on a fixed vector. We havesup X | φ v T ( x ) | = T sup X | φ π ( a T ) α ( x ) | = T sup X | φ α ( x · a T ) | = T sup X | φ α ( x ) | = β (cid:48)(cid:48) T , since α is a fixed vector in a fixed automorphic cuspidal representation ( ν, V ), and theaction does not change the supremum norm. (cid:3) Remark. It is easy to see that the condition that forms (cid:96) φ and (cid:96) φ (cid:48) are of the same levelis not essential for the proof, as well as that these are the Hecke forms. In particular,under our normalization of the measure on X (cid:96) , we see that for a vector v ∈ V , L -norms || (cid:96) φ v || X (cid:96) = || (cid:96) φ v || X (cid:96) (cid:48) are equal if (cid:96) | (cid:96) (cid:48) (here we view the function (cid:96) φ v as both Γ ( (cid:96)N )-invariant function and as Γ ( (cid:96) (cid:48) N )-invariant function). Hence for two Maass forms φ and φ (cid:48) on Γ ( N ), we obtain the bound: (cid:88) | λ i |≤ T |(cid:104) (cid:96) φ · (cid:96) φ (cid:48) , φ i (cid:105) Y (cid:96) (cid:48) N | · | λ i | e π | λ i | ≤ A · T , (10.12)where the summation is over an orthonormal basis of Maass forms for the subgroupΓ ( (cid:96) (cid:48) N ) with (cid:96) (cid:48) = lcm( (cid:96) , (cid:96) ). We also have the analogous bound for the full spectralexpansion (10.8) including the Eisenstein series contribution. The constant A in (10.12)depends on N and on eigenvalues of forms φ and φ (cid:48) . Holomorphic forms. The approach given above is applicable to holomorphicforms as well. In principle, there are no serious changes needed as compared to the Maassforms case. The main difficulty is that we have to fill in the gap left in [4] concerning themodel trilinear functional for the discrete series representations of P GL ( R ).Let φ k , φ (cid:48) k be (primitive) holomorphic forms of weight k for the subgroup Γ ( N ). Weassume these are L -normalized. For (cid:96) > 1, we consider (old) forms (cid:96) φ k ( z ) = φ k ( (cid:96)z )and (cid:96) φ (cid:48) k = φ (cid:48) k ( (cid:96)z ) on Γ ( (cid:96)N ). Under our normalization of measures for Y (cid:96)N , we have || (cid:96) φ k || Y (cid:96)N = || (cid:96) φ (cid:48) k || Y (cid:96)N = (cid:96) − k . This follows from the Rankin-Selberg method. Hence itwould have been more natural to consider normalized forms φ k | [ a (cid:96) ] k = (cid:96) k · (cid:96) φ k .For a (norm one) Maass form φ i on Γ ( (cid:96)N ), we define the corresponding triple coefficientby c ki ( (cid:96) ) = (cid:90) Y (cid:96)N (cid:96) φ k (cid:96) φ (cid:48) k φ i y k dv Y (cid:96)N . (10.13)As with Maass forms, we renormalize these coefficients in accordance with the Watsonformula by introducing normalized triple product coefficients | a ki ( (cid:96) ) | = | λ i | − k exp (cid:16) π | λ i | (cid:17) | c ki ( (cid:96) ) | . (10.14) Theorem B. There exists an effectively computable constant A such that the followingbound holds for all T ≥ , (cid:88) T ≤| λ i |≤ T | a ki ( (cid:96) ) | ≤ A · (cid:96) − k T , (10.15) where the summation is over an orthonormal basis of Maass forms for the group Γ ( (cid:96)N ) .The constant A depends on N , φ and φ (cid:48) , but not on (cid:96) .Remark. The proof we give applies to a slightly more general setup of forms of differentlevel (cid:96) and (cid:96) . Namely, we have (cid:88) T ≤| λ i |≤ T |(cid:104) (cid:96) φ k (cid:96) φ (cid:48) k , φ i (cid:105) Y (cid:96) (cid:48) N | · ( (cid:96) (cid:96) ) k | λ i | − k e π | λ i | ≤ A · T , (10.16)for two fixed forms φ k and φ (cid:48) k on Γ ( N ) and for (cid:96) (cid:48) = lcm( (cid:96) , (cid:96) ). Breaking the interval[1 , T ] into dyadic parts, we obtain for the full range (with the Eisenstein series contributionincluded), (cid:88) | λ i |≤ T |(cid:104) (cid:96) φ k (cid:96) φ (cid:48) k , φ i (cid:105) Y (cid:96) (cid:48) N | · ( (cid:96) (cid:96) ) k | λ i | − k e π | λ i | + (10.17) (cid:88) { κ } (cid:90) T − T (cid:104) (cid:96) φ k (cid:96) φ (cid:48) k , E κ ( it ) (cid:105) Y (cid:96) (cid:48) N | · ( (cid:96) (cid:96) ) k | t | − k e π | t | dt ≤ A · T ln( T ) . This is slightly weaker than (10.7) for Maass forms. ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 65 Proof of Theorem B. As we seen in the case of Maass forms, the proof is basedon the explicit form of the trilinear functional, its value on special vectors leading tothe normalization (10.14), and the construction of test vectors for which we can estimatesupremum norm effectively. We explain below changes and additions needed in order tocarry out this scheme for discrete series.10.5.1. Discrete series. Let k ≥ D k , π D k ) be the correspondingdiscrete series representation of P GL ( R ). In particular, for m ∈ Z , the space of K -typesof weight m is non-zero (and in this case is one-dimensional) if and only if | m | ≥ k . Thisdefines π k uniquely. Under the restriction to P SL ( R ), the representation π k splits intotwo representations ( D ± k , π ± D k ) of “holomorphic” and “anti-holomorphic” discrete series of P SL ( R ), and the element δ = diag(1 , − 1) interchanges them.We consider two realizations of discrete series as subrepresentations and as quotients ofinduced representations. Consider the space H k − of smooth even homogeneous functionson R \ k − f ( tx ) = t k − f ( x ) for any t ∈ R × and 0 (cid:54) = x ∈ R ). We have the natural action of GL ( R ) given by π k − ( g ) f ( x ) = f ( g − x ) · det( g ) ( k − / ,which is trivial on the center and hence defines a representation ( H k − , π k − ) of P GL ( R ).There exists a unique non-trivial invariant subspace W k − ⊂ H k − . The space W k − isfinite-dimensional, dim W k − = k − 1, and is generated by monomials x m x n , m + n = k − H k − /W k − is isomorphic to the space of smooth vectors of the discreteseries representation π k .We also consider the dual situation. Let H − k be the space of smooth even homogeneousfunctions on R \ − k . There is a natural P GL ( R )-invariantpairing (cid:104) , (cid:105) : H k − ⊗H − k → C given by the integration over S ⊂ R \ 0. Hence H − k is thesmooth dual of H k − , and vice versa. There exists a unique non-trivial invariant subspace D ∗ k ⊂ H − k . The quotient H − k /D ∗ k is isomorphic to the finite-dimensional representation W k − .Of course D ∗ k is isomorphic to D k , but we will distinguish between two realizations ofthe same abstract representation as a subrepresentation D ∗ k ⊂ H − k and as a quotient H k − → D k . We denote corresponding maps by i k : D ∗ k ⊂ H − k and q k : H k − → D k .10.5.2. Trilinear invariant functionals. Let ( V λ,(cid:15) , π λ,(cid:15) ) be a unitary representation of theprincipal series of P GL ( R ). These are parameterized by λ ∈ i R and by (cid:15) = 0 , δ (see [9]). The space Hom G ( D k ⊗ D ∗ k , V λ,(cid:15) ) is one-dimensional.We will work with the space of invariant trilinear functionals Hom G ( D k ⊗ D ∗ k ⊗ V − λ,(cid:15) , C )instead. We construct below a non-zero functional l indk,λ,(cid:15) ∈ Hom G ( H k − ⊗H − k ⊗ V − λ,(cid:15) , C ) forinduced representations (in fact, this space is also one-dimensional) by means of (analyticcontinuation of) an explicit kernel. We use it to define a non-zero functional l modk,λ,(cid:15) ∈ Hom G ( D k ⊗ D ∗ k ⊗ V − λ,(cid:15) , C ). What is more important, we will use l indk,τ,(cid:15) in order to carryout our computations in a way similar to the principal series. Let l indk,λ,(cid:15) ∈ Hom G ( H k − ⊗ H − k ⊗ V − λ,(cid:15) , C ) be a non-zero invariant functional. Such afunctional induces the corresponding functional on H k − ⊗ D ∗ k ⊗ V − λ,(cid:15) since D ∗ k ⊂ H − k .Moreover, any such functional vanishes on the subspace W k − ⊗ D ∗ k ⊗ V − λ,(cid:15) since thereare no non-zero maps between W k − ⊗ D ∗ k and V λ,(cid:15) . Hence we obtain a functional l modk,λ,(cid:15) ∈ Hom G ( D k ⊗ D ∗ k ⊗ V − λ,(cid:15) , C ) on the corresponding quotient space. We denote by T modk,λ,(cid:15) : D k ⊗ D ∗ k → V λ,(cid:15) the associated map, and by H modk,λ,(cid:15) ( u ) = || T modk,λ,(cid:15) ( u ) || V λ,(cid:15) , u ∈ D k ⊗ D ∗ k thecorresponding Hermitian form.10.5.3. Model functionals. We follow the construction from [4]. Denote K k,λ ( x, y, z ) = | x − y | − − λ | x − z | − λ − k +1 | y − z | − λ + k − . (10.18)In order to construct l modk,λ,(cid:15) ∈ Hom G ( H k − ⊗ H − k ⊗ V − λ,(cid:15) , C ), we consider the followingfunction in three variables x, y, z ∈ R K k − , − k,λ,(cid:15) ( x, y, z ) = ( sgn ( x, y, z )) (cid:15) · K k,λ ( x, y, z ) , (10.19)where sgn ( x , x , z ) = (cid:81) i (cid:54) = j sgn ( x i − x j ) (this is an SL ( R )-invariant function on R distinguishing two open orbits). An analogous expression could be written in the circlemodel on the space C ∞ ( S ). Viewed as a kernel, K k − , − k,λ,(cid:15) defines an invariant non-zerofunctional l indk,λ,(cid:15) on the (smooth part of) the representation H k − ⊗ H − k ⊗ V − τ,(cid:15) ⊂ C ∞ ( R ).Such a kernel should be understood in the regularized sense (e.g., analytically continuedfollowing [16]). We are interested in λ ∈ i R , | λ | → ∞ , and hence all exponents in (10.19)are non integral. This implies that the regularized kernel does not have a pole at relevantpoints.We denoted by l modk,λ,(cid:15) ∈ Hom G ( D k ⊗ D ∗ k ⊗ V − λ,(cid:15) , C ) the corresponding model functional.The difference with principal series clearly lies in the fact that we only can compute theauxiliary functional l indk,λ,(cid:15) . However, for k fixed, it turns out that necessary computationsare essentially identical to the ones we performed for the principal series in [4].10.5.4. Value on K -types. In order to obtain the normalization (10.14) and to compareour model functional l modk,λ,(cid:15) to the automorphic triple product (10.13), we have to compute,or at least to bound, the value l modk,λ,(cid:15) ( e k ⊗ e − k ⊗ e ) where e ± k ∈ D k are highest / lowest K -types of norm one, and e ∈ V λ,(cid:15) is a K -fixed vector of norm one. For Maass forms,this is done in the Appendix of [4] by explicitly calculating this value in terms of Γ-functions. In fact, the relevant calculation is valid for K -fixed vectors for any threeinduced representations with generic values of parameters (i.e., those for which the finalexpression is well-defined). Using the action of the Lie algebra of G (see [37] for thecorresponding calculation where it is used to prove uniqueness), one obtains recurrencerelations between values of the model functional on various weight vectors. For a genericvalue of τ , this allows one to reduce the computation of l modτ,λ,(cid:15) ( e k ⊗ e − k ⊗ e ) to the value of l modτ,λ,(cid:15) ( e ⊗ e ⊗ e ). By analytic continuation, this relation holds for our set of parameters ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 67 corresponding to the discrete series. From this, one deduces the bound | l modk,λ,(cid:15) ( e k ⊗ e − k ⊗ e ) | ≤ a | λ i | k − exp (cid:16) − π | λ i | (cid:17) , (10.20)for some explicit constant a > k . In fact, this is the actual order of themagnitude for the above value, and one can obtain the exact value in terms of the Eulergamma function and that way derive the asymptotic for the value l modk,λ,(cid:15) ( e k ⊗ e − k ⊗ e ) as | λ | → ∞ . Remark. There is a natural trilinear functional on Whittaker models of representationsof G . This is the model which appears in the Rankin-Selberg method as a result ofunfolding. The above computation (and the similar one for Maass forms performed in[4]) shows that our normalization of the trilinear functional and the one coming from theWhittaker model coincide up to a constant of the absolute value one.10.5.5. Test vectors. Our construction is very close to the construction we made in Sec-tion 10.3.1 for principal series representations, with appropriate modifications. We con-struct a test vector u T ( x, y ) = v T ( x ) ⊗ v (cid:48) T ( y ) ∈ D k ⊗ D ∗ k ⊂ H k − ⊗ H − k satisfying H modk,λ,(cid:15) ( u T ) ≥ β > T / ≤ | λ | ≤ T and some constant β > λ . Vectors v T ∈ H k − and v (cid:48) T ∈ H − k are first constructed in the line model of induced representations,and then we relate these to vectors in the discrete series representation D k .Choose a smooth non-negative function α ∈ C ∞ ( R ) with the support supp( α ) ⊂ [ − ι, ι ]for ι > (cid:82) R α ( x ) dx = 1. Considerthe diagonal element a T = diag( T − , T ) ∈ G . We define w T ∈ H k − by w T = T · π k − ( a T ) α . (10.21)Recall that the action is given by π k − (diag( a − , a )) v ( x ) = | a | − k v ( a x ). We note thatgeometrically the vector w T is a small non-negative bump function around the point0 ∈ R , with the support in the interval T − · [ − ι, ι ], and satisfying (cid:82) R w T ( x ) dx = T (1 − k ) .We now set v T = q k ( w T ) ∈ D k , where q k : H k − → D k is the projection. Note that v T = T · π D k ( a T )˜ v for some ˜ v ∈ D k , i.e., the vector v T is obtained by the action of G onsome fixed vector in D k . This will be crucial in what follows since we will need to estimatethe supremum norm for the automorphic realization of the vector v T .The construction of the test vector v (cid:48) T ∈ D ∗ k is slightly more complicated since we cannotsimply project a vector to D ∗ k ⊂ H − k because the value of the functional l indk,λ,(cid:15) mightchange significantly. Let α be as above. We now view it as a vector in H − k . We choosea smooth real valued function α (cid:48) ∈ C ∞ ( R ) satisfying the following properties:(1) supp( α (cid:48) ) ⊂ [ M − ι, M + ι ], where the parameter M is to be chosen later,(2) (cid:82) R x m α (cid:48) ( x ) dx = − (cid:82) R x m α ( x ) dx for 0 ≤ m ≤ k − . The last condition implies that the vector w = α + α (cid:48) ∈ D ∗ k since (cid:82) R x m w ( x ) dx = 0 for0 ≤ m ≤ k − 2. We define now the second test vector by v (cid:48) T = T · π − k (( ) a T ) w . (10.22)Clearly we have v (cid:48) T ∈ D ∗ k .Recall that π − k (diag( a − , a )) v ( x ) = | a | k v ( a x ), and π − k (( )) v ( x ) = v ( x − v (cid:48) T = α T + α (cid:48) T is the sum of two bump functions α T and α (cid:48) T withtheir supports satisfying supp( α T ) ⊂ T − [ − ι, ι ] and supp( α (cid:48) T ) ⊂ T − [ M − ι, M + ι ],both near the point 1 ∈ R . We also have α T ≥ (cid:82) R α T ( x ) dx = T ( k − .We now set our test vector to u T = v T ⊗ v (cid:48) T ∈ D k ⊗ D ∗ k . We want to show that for T / ≤ | λ | ≤ T , | l modk,λ,(cid:15) ( u T ⊗ u ) | ≥ c (cid:48) > u ∈ V λ,(cid:15) with || u || = 1, and with a constant c (cid:48) > λ .As we explained before, l modk,λ,(cid:15) ( q k ( v ) ⊗ w ⊗ u ) = l indk,λ,(cid:15) ( v ⊗ w ⊗ u ) for any triple v ⊗ w ⊗ u ∈H k − ⊗ H − k ⊗ V λ,(cid:15) . Hence we work with l indk,λ,(cid:15) ( w T ⊗ v (cid:48) T ⊗ u ) instead of l modk,λ,(cid:15) because l indk,λ,(cid:15) is given by an explicit integral. Let K k,λ ( x, y, z ) be as in (10.18). We have then l indk,λ,(cid:15) ( w T ⊗ v (cid:48) T ⊗ u ) = (cid:90) K k,λ ( x, y, z )( sgn ( x, y, z )) (cid:15) w T ( x ) v (cid:48) T ( y ) u ( z ) dxdydz . (10.24)Hence it is enough to show that the absolute value of the integral I λ ( z ) = (cid:90) K k,λ ( x, y, z ) w T ( x ) v (cid:48) T ( y ) dxdy = (cid:104) K k,λ ( x, y, z ) , w T ( x ) v (cid:48) T ( y ) (cid:105) (10.25)is not small for z in some fixed interval of R .We have I λ ( z ) = K ( z ) + K (cid:48) ( z ), where K ( z ) = (cid:104) K k,λ ( x, y, z ) , w T ( x ) α T ( y ) (cid:105) and K (cid:48) ( z ) = (cid:104) K k,λ ( x, y, z ) , w T ( x ) α (cid:48) T ( y ) (cid:105) . We will show now that for a certain range of z , integrands inthese integrals are not small, do not oscillate, and that integrals do not cancel each other.This is implies the existence of a vector u above (e.g., L -normalized smooth characteristicfunction supported in the same range of z ).The function w T ( x ) α T ( y ) is a non-negative function with the support in a small box ofsize ιT − around the point (0 , ∈ R , and the gradient of the function K k,λ ( x, y, z )is bounded by | λ | ≤ T in this box. This implies that for a small enough (but fixed) ι > 0, the argument of K k,λ ( x, y, z ) belongs to a small interval of S for x, y, z satisfyingthe above restrictions. It is also easy to see that there exists a constant c (cid:48)(cid:48) > | K ( z ) | ≥ c (cid:48)(cid:48) > z ∈ [10 , 20] and | λ | ≤ T . Hence there are no significantcancellations in the integral (10.25) for z in the above range. We normalized our vectorsso that (cid:82) R v T ( x ) α T ( y ) dxdy = 1, and hence the integral K ( z ) is not small for z not nearsingularities of the kernel K k,λ ( x, y, z ) which is the case because of the restriction on x and y . This part is identical to our argument in Section 5.3.4 of [4]. ULTIPLE DIRICHLET SERIES AND SHIFTED CONVOLUTIONS 69 We are left with the second term K (cid:48) ( z ). We want to show that there are no cancellationsbetween two terms K ( z ) and K (cid:48) ( z ) for z ∈ [10 , λ isnot too small (e.g., T / ≤ | λ | ≤ T ). Namely, the argument of the kernel function K k,λ in(10.25) on the support of u T is given by λ/ T · (cid:2) t (1 − z − ) − t (1 + ( z − − ) (cid:3) + λ/ z ) + ln( z − O ( T − ) , (10.26)where t ∈ [ − ι, ι ], t ∈ [1 − ι, ι ] for α T , and t ∈ [ M − ι, M + ι ] for α (cid:48) T . By choosingappropriate value of M , we can see that the difference of these arguments is not close to0 and π for any fixed z ∈ [10 , 20] and T / ≤ | λ | ≤ T . Hence integrals K ( z ) and K (cid:48) ( z ) donot cancel each other since u T is real valued.We have shown that H λ ( u T ) ≥ c (cid:48) > T / ≤ | λ | ≤ T , and some explicit c (cid:48) > λ .10.5.6. Raising the level. We now discuss what happens when we change the level. Sinceour test vectors v T and v (cid:48) T are not K -finite, we have to pass to automorphic functions onthe space X (cid:96) . We use the standard notation j ( g, z ) = det( g ) − ( cz + d ) for g = ( a bc d ) ∈ G + and z ∈ H . Let φ k be a primitive holomorphic form of weight k on H for the subgroupΓ ( N ). We normalize φ k by its norm on Y N . According to the well-known dictionary,we associate to φ k the function φ e k ∈ C ∞ ( X ) given by φ e k ( g ) = φ k ( g ( i )) · j ( g, z ) − k , (10.27)where z = g ( i ). In the opposite direction, we have φ k ( g ( i )) = φ e k ( g ) · j ( g, z ) k . We havethe associated isometry ν k = ν φ k : D k → C ∞ ( X N ) which gives ν k ( e k ) = φ e k .Let (cid:96) > a (cid:96) = diag( (cid:96) , (cid:96) − ). For a given ν k = ν φ k : D k → C ∞ ( X N ), we construct the corresponding isometry ν (cid:96)k : D k → C ∞ ( X (cid:96) ) as follows. For avector v ∈ D k , we consider the corresponding automorphic function (cid:96) φ v ( x ) = ν (cid:96)k ( v )( x ) = ν k ( v ) ( a (cid:96) x ) = φ v ( a (cid:96) x ) . (10.28)Obviously, we have sup X (cid:96) | (cid:96) φ v | = sup X | φ v | for the same vector v ∈ D k . We want tocompare this to the classical normalization of old forms. For the lowest weight vector e k ∈ D k , we have with z = g ( i ) (cid:96) φ e k ( g ) = φ e k ( a (cid:96) g ) = φ k ( (cid:96)z ) · j ( a (cid:96) g, i ) − k = (cid:96) k φ k ( (cid:96)z ) · j ( g, i ) − k . (10.29)On the other hand, classically, old forms are given by (cid:96) φ k ( z ) = φ k ( (cid:96)z ). Hence we acquirethe extra factor (cid:96) k .Since, as we noted, test vectors v T ∈ D k and v (cid:48) T ∈ D ∗ k are obtained by the (scaled) groupaction applied to fixed vectors, and since the operation of raising the level by (cid:96) does notchange the supremum norm, we arrive at the following boundsup X (cid:96) | (cid:96) φ v T ( x ) | = sup X | φ v T ( x ) | = T sup X | φ π ( a T ) w ( x ) | = T sup X | φ w ( x ) | = β (cid:48) T , for some constant β (cid:48) . The same holds for the automorphic function (cid:96) φ v (cid:48) T . This impliesthat || (cid:96) φ v T (cid:96) φ v (cid:48) T || X (cid:96) ≤ βT for some β > (cid:96) and T .To summarize, we have proved the bound (cid:88) T ≤| λ i |≤ T |(cid:104) (cid:96) φ k (cid:96) φ (cid:48) k , φ i (cid:105) Y (cid:96)N | · (cid:96) k | λ i | − k e π | λ i | ≤ A · T , (10.30)where the summation is over an orthonormal basis of Maass forms for the subgroupΓ ( (cid:96)N ). The constant A in (10.30) depends on N and on the weight k of forms φ and φ (cid:48) .The above argument also proves the case of forms with different level, i.e., the bound(10.16). Under the normalization of measures on X (cid:96) , we see that for a vector v ∈ V , L -norms || (cid:96) φ v || X (cid:96) = || (cid:96) φ v || X (cid:96) (cid:48) are equal if (cid:96) | (cid:96) (cid:48) (here we view the function (cid:96) φ v as bothΓ ( (cid:96)N )-invariant function and as Γ ( (cid:96) (cid:48) N )-invariant function). Obviously, the supremumnorms of (cid:96) φ v on X (cid:96) and on X (cid:96) (cid:48) are also coincide. Hence for integers (cid:96) and (cid:96) , and (cid:96) (cid:48) = lcm( (cid:96) , (cid:96) ), we have || (cid:96) φ v T (cid:96) φ v (cid:48) T || X (cid:96) (cid:48) ≤ || (cid:96) φ v T || X (cid:96) (cid:48) + 12 || (cid:96) φ v (cid:48) T || X (cid:96) (cid:48) ≤ βT . This implies (10.16). (cid:3) Acknowledgments. 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