The Second Moment of Sums of Coefficients of Cusp Forms
Thomas A. Hulse, Chan Ieong Kuan, David Lowry-Duda, Alexander Walker
aa r X i v : . [ m a t h . N T ] N ov THE SECOND MOMENT OF SUMS OF COEFFICIENTSOF CUSP FORMS
THOMAS A. HULSE, CHAN IEONG KUAN, DAVID LOWRY-DUDA,AND ALEXANDER WALKER
Abstract.
Let f and g be weight k holomorphic cusp forms and let S f ( n ) and S g ( n ) denote the sums of their first n Fourier coefficients.Hafner and Ivi´c [HI89] proved asymptotics for P n ≤ X | S f ( n ) | and provedthat the Classical Conjecture, that S f ( X ) ≪ X k − + + ǫ , holds on av-erage over long intervals.In this paper, we introduce and obtain meromorphic continuationsfor the Dirichlet series D ( s, S f × S g ) = P S f ( n ) S g ( n ) n − ( s + k − and D ( s, S f × S g ) = P n S f ( n ) S g ( n ) n − ( s + k − . We then prove asymptoticsfor the smoothed second moment sums P S f ( n ) S g ( n ) e − n/X , giving asmoothed generalization of [HI89]. We also attain asymptotics for anal-ogous sums of normalized Fourier coefficients. Our methodology extendsto a wide variety of weights and levels, and comparison with [CN62] in-dicates very general cancellation between the Rankin-Selberg L -function L ( s, f × g ) and convolution sums of the coefficients of f and g .In forthcoming works, the authors apply the results of this paper toprove the Classical Conjecture on | S f ( n ) | is true on short intervals, andto prove sign change results on { S f ( n ) } n ∈ N . Introduction and Statement of Results
Let f be a holomorphic cusp form on a congruence subgroup Γ ⊆ SL ( Z )and of positive weight k , where k ∈ Z ∪ ( Z + ). Let the Fourier expansionof f at ∞ be given by f ( z ) = X n ≥ a ( n ) e ( nz ) , where e ( z ) = e πiz . In this paper, we consider upper bounds for the secondmoment of the partial sums of the Fourier coefficients, S f ( n ) := X m ≤ n a ( m ) . Bounds on the coefficients a ( n ) are of great interest and have wide ap-plication. The famous Ramanujan-Petersson conjecture, which was proven Research of the first author was supported by a Coleman Postdoctoral Fellowship atQueen’s University.The third author is supported by the National Science Foundation Graduate ResearchFellowship Program under Grant No. DGE 0228243. to hold for integral weight holomorphic cusp forms as a consequence ofDeligne’s proof of the Weil Conjecture [Del74], gives us that a ( n ) ≪ n k − + ǫ and from this one might naively assume S f ( X ) ≪ X k − +1+ ǫ . However,there is significant cancellation in the sum and we expect the far betterbound, S f ( X ) ≪ X k − + + ǫ , (1.1)which we refer to as the “Classical Conjecture,” echoing Hafner and Ivi´c intheir work [HI89].Chandrasekharan and Narasimhan, as a consequence of their much broaderwork on the average order of arithmetical functions [CN64], [CN62], provedthat the Classical Conjecture is true on average by showing that X n ≤ X | S f ( n ) | = CX k − + B ( X ) , (1.2)where B ( x ) is an error term, B ( X ) = ( O ( X k log ( X ))Ω (cid:16) X k − (log log log X ) log X (cid:17) , (1.3)and C is the constant, C = 1(4 k + 2) π X n ≥ | a ( n ) | n k + . (1.4)A application of the Cauchy-Schwarz inequality to (1.2) leads to the on-average statement that 1 X X n ≤ X | S f ( n ) | ≪ X k − + . (1.5)From this, Hafner and Ivi´c [HI89] were able to show that for holomorphiccusp forms with real coefficients of full integral weight on SL ( Z ), S f ( X ) ≪ X k − + . (1.6)Their argument requires cusp forms of full level and the Ramanujan-PeterssonConjecture, but one can state analogous results for general level in terms ofthe corresponding best-known progress towards the conjecture.Better lower bounds are known for B ( X ). In the same work, [HI89],Hafner and Ivi´c improved the lower bound of Chandrasekharan and Narasimhanfor full-integral weight forms of level one and showed that B ( X ) = Ω (cid:16) X k − exp (cid:16) D (log log x ) / (log log log x ) / (cid:17)(cid:17) , (1.7)for a particular constant D .In general, it is possible to translate upper bounds for B ( X ) into up-per bounds for S f ( X ). By the Ramanujan-Petersson Conjecture, we have | S f ( X + ℓ ) | ≫ | S f ( X ) | for all ℓ ≪ α ( X ) where α ( X ) := | S f ( X ) | X − k − − ǫ , UMS OF FOURIER COEFFICIENTS 3 for any small ǫ >
0, and X sufficiently large. It follows from (1.2), and thebinomial expansion of ( X ± α ( X )) k + that | S f ( X ) | X k − + ǫ ≪ X | n − X |≤ α ( X ) | S f ( n ) | = O (cid:16) X k − + − ǫ | S f ( X ) | + B ( X ) (cid:17) , so that | S f ( X ) | ≪ X k − + or | S f ( X ) | ≪ B ( X ) X k − . The first possibil-ity is the Classical Conjecture and the second relates B ( X ) to S f ( X ). Inparticular, the bound B ( X ) ≪ X k log ( X ) from [CN62] nearly allows us torecover Hafner and Ivi´c’s upper bound (1.6).In this paper, we consider Dirichlet series associated to the norm-squaredpartial sums, | S f ( n ) | , and squared partial sums, S f ( n ) , D ( s, S f × S f ) := X n ≥ | S f ( n ) | n s + k − , D ( s, S f × S f ) := X n ≥ S f ( n ) n s + k − . (1.8)The extra factor of n k − appearing in the denominator serves to normalizethe L -functions and shifted convolution sums that appear in the decompo-sition of these series, as we will see in Section 3, so that every functionalequation is of the form s − s . We choose our notation to be reminiscentof the notation for the Rankin-Selberg convolution. We have been unable tofind previous occurrences of the above series in the literature, but sufficientunderstanding of the analytic properties of D ( s, S f × S f ) and D ( s, S f × S f )allows new insights and avenues for investigating the behaviour of S f ( n ).More generally, we also study the Dirichlet series associated to the productof the partial sums of two weight k holomorphic cusp forms, f and g , D ( s, S f × S g ) := X n ≥ S f ( n ) S g ( n ) n s + k − , D ( s, S f × S g ) := X n ≥ S f ( n ) S g ( n ) n s + k − . (1.9)Initially we work only with f, g of full-integral weight and of level one. Themethodology is almost identical for D ( s, S f × S g ) as it is for D ( s, S f × S g ),so we exposit only for D ( s, S f × S g ) and state the results for D ( s, S f × S g ).In Section 3, we decompose D ( s, S f × S g ) into pieces which we refer to asthe diagonal and off-diagonal contributions. Our main result is the mero-morphic continuation of these diagonal and off-diagonal pieces, given byTheorem 4.7 in Section 4, ultimately giving the meromorphic continuationsof D ( s, S f × S g ) and D ( s, S f × S g ).The major difficulty arises in determining the analytic behaviour of the off-diagonal , which involves the shifted convolution sum Z ( s, w, f × g ) := X n,h ≥ a ( n ) b ( n − h ) + a ( n − h ) b ( n ) n s + k − h w , where the a ( n ) and b ( n ) are coefficients of f and g , respectively. Our ap-proach and notation to determine the analytic behaviour of Z ( s, w, f × g ) HULSE, KUAN, LOWRY-DUDA, AND WALKER is similar to that in [HHR13], though the technique originates with Sel-berg [Sel65]. In particular, the above symmetrized form of Z ( s, w, f × g )is reminiscent of a construction devised by Bringmen, Mertens and Ono in[BMO16]. In Section 4, we describe the behaviour of Z ( s, , f × f ) and showthat there is remarkable cancellation at potential poles, both within thesum in its spectral expansion and with the pole due to the Rankin-Selberg L -function L ( s, f × f ).We use an inverse Mellin transform and apply the meromorphic continu-ation of D ( s, S f × S g ) in Section 5 to obtain the following smoothed partialsum result. Theorem 1.1.
Suppose that f and g are holomorphic cusp forms of SL ( Z ) of integral weight k . Then for any ǫ > , X X n ≥ S f ( n ) S g ( n ) n k − e − n/X = CX + O f,g,ǫ ( X − + θ + ǫ ) (1.10) and X X n ≥ S f ( n ) S g ( n ) n k − e − n/X = C ′ X + O f,g,ǫ ( X − + θ + ǫ ) , (1.11) where C = Γ( )4 π L ( , f × g ) ζ (3) = Γ( )4 π X n ≥ a ( n ) b ( n ) n k + , and C ′ = Γ( )4 π L ( , f × g ) ζ (3) = Γ( )4 π X n ≥ a ( n ) b ( n ) n k + , with θ = max j (Im t j ) , where + it j correspond to the types of each form inan orthogonal basis of Maass cusp forms. (For SL ( Z ) we know that θ = 0 .) Remark 1.2.
Our main proof of Theorem 1.1 is for the case when f and g are level one. We expect the argument to completely generalize to higherlevel and nebentypus, and thus also to half-integral weight, but it is moredifficult to explicitly verify the absence of poles of D ( s, S f × S g ) and D ( s, S f × S g ) in this case. We reserve the details concerning general level for Section 6.This can be thought of as a generalized, smoothed analogue of (1.2), asit includes a weight factor of e − n/X rather than a sharp cutoff function. Inparticular, by specializing to eigenforms f = g , we recover a new proof ofthe Classical Conjecture on average. The disparity between the error terms, O ( X − + θ + ǫ ) in our theorem and O ( X k log ( X )), as cited above in (1.3), isaccounted for in the smoothing factor. Applying different Mellin transformswould allow us to obtain sharp cutoff results but by our current analysistechniques we are unable to match O ( X k log ( X )), which we still expect tobe true even in the case when f = g . Indeed, the bound on the error term UMS OF FOURIER COEFFICIENTS 5 B ( x ) in (1.3) does not come from a pole of D ( s, S f × S f ), but rather frommore subtle issues of convergence from using a sharp cutoff.In Section 6, we show how our methods might extend to general level anddemonstrate how they extend to square-free level explicitly. Conversely, bycomparing our methodology with the results of [HI89] and [CN62], we getTheorem 6.2, which demonstrates surprising cancellation involving Kloost-erman sums for general level. Areas for Further Investigation.
By using either a sharp cutoff or aconcentrating integral transform on the meromorphic continuation given inTheorems 4.7 one could hope to extend the methods of this paper to getindividual bounds on S f ( n ) instead of average bounds. For the first time,we have an analytic object with which to explore these second moments.The greatest barrier to progress has been understanding the discrete spec-trum (4.7), which requires bounding12 πi Z ( σ ) X j ρ j (1) Γ( s − + it j )Γ( s − − it j )Γ( s )Γ( s + k − L ( s − , µ j ) × h Im( · ) k ( f g + T − ( f g )) , µ j i V ( X, s ) ds where V ( X, s ) is a cutoff function and T − is the Hecke operator that sends x in the argument of the function to − x . Other choices of cutoff functionshould lead to more and different avenues for exploration. We suspect bet-ter understanding of bounds on terms coming from the discrete spectrumare possible with current technology, and we are eager to explore other ap-proaches.Our current ability to bound the discrete spectrum will be made moreclear in forthcoming work, where we build on the meromorphic properties of D ( s, S f × S f ) and use the flexible cutoff function V Y ( X, s ) = X Y exp( πs /Y )to bound the size of S f ( n ) in short intervals of a particular size.Further, bounds on the size of P S f ( n ) are closely related to sign changesof S f ( n ) and of the individual Fourier coefficients a ( n ). In another forthcom-ing work, the authors investigate the sign changes in the sequence { S f ( n ) } and relations with the Classical Conjecture.2. Basic Tools and Notation
In this section, we recall some basic tools and formulae.2.1.
The Rankin-Selberg L -function. Here we follow the constructionof L ( s, f × g ) given in Section 1.6 of [Bum98] but we normalize so thatthe functional equation corresponds to the transformation s → − s . Let f ( z ) = P a ( n ) e ( nz ) and g ( z ) = P b ( n ) e ( nz ) be modular forms of weight k on a congruence subgroup Γ, and where at least one is cuspidal. Let Γ \H denote the upper half plane modulo the group action due to Γ and let h f, g i HULSE, KUAN, LOWRY-DUDA, AND WALKER denote the Petersson inner product, h f, g i = Z Z Γ \H f ( z ) g ( z ) dxdyy . The Rankin-Selberg convolution L -function is given by the Dirichlet seriesexpansion for Re s > L ( s, f × g ) := ζ (2 s ) X n ≥ a ( n ) b ( n ) n s + k − , (2.1)which can be meromorphically continued to all s ∈ C via the identity, L ( s, f × g ) = (4 π ) s + k − ζ (2 s )Γ( s + k − D Im( · ) k f g, E ( · , s ) E , (2.2)where E ( z, s ) is the real-analytic Eisenstein series E ( z, s ) = X γ ∈ Γ ∞ \ Γ Im( γz ) s , (2.3)for Re s >
1. We also note that in (2.2) if we replace f g with f T − g , where T − is the Hecke operator which has action T − F ( x + iy ) = F ( − x + iy ) asdescribed in Theorem 3.12.6 of [Gol06], we can similarly get a meromorphiccontinuation of L ( s, f × g ). Here Γ ∞ is the stabilizer subgroup of Γ of thecusp at infinity. The Rankin-Selberg L -function is holomorphic except for,at most, a simple pole at s = 1 when f = g whose residues can be read from(2.2). When Γ = SL ( Z ), we have the functional equation(2 π ) − s Γ( s )Γ( s + k − L ( s, f × g ) := Λ( s, f × g ) = Λ(1 − s, f × g )due to the functional equation of the completed Eisenstein series E ∗ ( z, s ) = E ∗ ( z, − s ) where E ∗ ( z, s ) := π − s Γ( s ) ζ (2 s ) E ( z, s ). There is an analogoustransformation at higher levels but its formulation is complicated by theexistence of other cusps.2.2. Mellin-Barnes integral transform.
Here we consider an integralanalogue of the binomial theorem and one of the integrals considered byBarnes [Bar08], slightly modified from the formulation presented in 6.422(3)of [JZ07].
Lemma 2.1 (Barnes, 1908) . If > γ > − Re( β ) and | arg t | < π , then πi Z γ + i ∞ γ − i ∞ Γ( − s )Γ( β + s ) t s ds = Γ( β )(1 + t ) − β . UMS OF FOURIER COEFFICIENTS 7
Selberg spectral expansion.
We review Selberg’s Spectral Theorem,as presented in Theorem 15.5 of [IK04].Let L (Γ \H ) denote the space of square integrable functions with respectto the Petersson norm. Let f ∈ L (Γ \H ) and let { µ j ( z ) : j ≥ } be acomplete orthonormal system for the residual and cuspidal spaces of Γ \H ,consisting of the constant function µ ( z ) and infinitely many Maass cuspforms µ j ( z ) for j ≥ + t j with respect tothe hyperbolic Laplacian. We may assume that µ j are also simultaneouseigenfunctions of the Hecke operators, including the T − operator. Then f ( z ) has the spectral decomposition, f ( z ) = X j h f, µ j i µ j ( z ) + X a π Z R h f, E a ( · , + it ) i E a ( z, + it ) dt, (2.4)where a ranges over the cusps of Γ \H . We casually refer to the first sumas the discrete and residual spectrum, and the sums of integrals as thecontinuous spectrum.Further, these it j of the Maass cusp forms µ j are expected to satisfy Sel-berg’s Eigenvalue Conjecture, which says that all t j are real. It is knownthat t j is purely real or purely imaginary, and Selberg’s Eigenvalue Conjec-ture has been proved for certain congruence subgroups, but not in general.We let θ = sup j {| Im( t j ) |} denote the best known progress toward Selberg’sEigenvalue Conjecture for Γ. The current best known result for θ for allcongruence subgroups is due to Kim and Sarnak, who show that θ ≤ asa consequence of a functoriality result due to Kim [Kim03].3. Decomposition of Main Series
In this section, we decompose D ( s, S f × S g ) into smaller components thatwe can analyze individually. Proposition 3.1.
Let f ( z ) = P n ≥ a ( n ) e ( nz ) and g ( z ) = P n ≥ b ( n ) e ( nz ) be two weight k cusp forms. Define S f ( n ) := P m ≤ n a ( m ) to be the partialsum of the first n coefficients of f . Then the Dirichlet series associated to S f ( n ) S g ( n ) decomposes into D ( s, S f × S g ) := X n ≥ S f ( n ) S g ( n ) n s + k − (3.1)= W ( s ; f, g ) + 12 πi Z ( γ ) W ( s − z ; f, g ) ζ ( z ) Γ( z )Γ( s − z + k − s + k − dz, for < γ < Re( s − . Here, L ( s, f × g ) denotes the Rankin-Selberg L -function as in Section 2.1, W ( s ; f, g ) is given by W ( s ; f, g ) := L ( s, f × g ) ζ (2 s ) + Z ( s, , f × g ) , (3.2) HULSE, KUAN, LOWRY-DUDA, AND WALKER and Z ( s, w, f × g ) denotes the symmetrized shifted convolution sum Z ( s, w, f × g ) := X n,h ≥ a ( n ) b ( n − h ) + a ( n − h ) b ( n ) n s + k − h w . (3.3) Proof.
Expand and recollect the partial sums S f and S g . D ( s, S f × S g ) = X n ≥ S f ( n ) S g ( n ) n s + k − = ∞ X n =1 n s + k − n X m =1 a ( m ) n X h =1 b ( h ) . (3.4)Separate the sums over m and h into the cases where m = h, m > h , and m < h . We adopt the convention that a ( n ) = 0 for n ≤ n instead of up to n . With somewhat abusive summation notation, this gives ∞ X n =1 n s + k − X h = m ≥ + X h>m ≥ + X m>h ≥ a ( n − m ) b ( n − h ) . (3.5)In the first above sum, we take h = m . In the second sum, when h > m , wecan let h = m + ℓ and then sum over m and ℓ . Similarly in the third sum,when m > h , we can let m = h + ℓ . This yields= ∞ X n =1 n s + k − (cid:18) X m ≥ a ( n − m ) b ( n − m )+ X ℓ ≥ m ≥ a ( n − m ) b ( n − m − ℓ ) + X ℓ ≥ m ≥ a ( n − m − ℓ ) b ( n − m ) (cid:19) . The cases when m = 0 are distinguished. They contribute W ( s ; f, g ) = L ( s, f × g ) ζ (2 s ) + Z ( s, , f × g ) . So W ( s ; f, g ) is a separation into diagonal, above-diagonal, and below-diagonalcomponents. After reindexing by changing n n + m , the sums over m ≥ X n,m ≥ n + m ) s + k − (cid:18) a ( n ) b ( n ) + X ℓ ≥ a ( n ) b ( n − ℓ ) + X ℓ ≥ a ( n − ℓ ) b ( n ) (cid:19) . By using the Mellin-Barnes transform from Section 2.2 with t = m/n ,we decouple m from n . Restricting to γ > s sufficiently large,the m sum can be collected into ζ ( z ) and the n sum can be collected into W ( s ; f, g ). Simplification completes the proof. (cid:3) To understand D ( s, S f × S g ), we study the analytic behaviour of L ( s, f × g )and Z ( s, w, f × g ). We treat W ( s ; f, g ) as a single object and we show inSection 4 that the pole of L ( s, f × g ) will exactly cancel the rightmost poleof Z ( s, , f × g ). UMS OF FOURIER COEFFICIENTS 9 Analytic behaviour of W ( s ; f, g ) and Z ( s, w, f × g )For now, let f and g be full-integral weight holomorphic cusp forms oflevel one. We expect most of our methods will generalize to arbitrary leveland to half-integral weight, and we try to present the material in a way thatindicates how the general methodology works. To this end, we continueto show dependence on progress towards Selberg’s Eigenvalue Conjectureto indicate how the results generalize to congruence subgroups where theconjecture is not yet verified. We return to more general level in Section 6.In this section, we produce a spectral expansion for the symmetrized shifteddouble Dirichlet series Z ( s, w, f × g ) := X m ≥ X ℓ ≥ a ( m ) b ( m − ℓ ) + a ( m − ℓ ) b ( m ) m s + k − ℓ w , and use it to understand the analytic behaviour of W ( s ; f, g ).4.1. Spectral Expansion.
For an integer h ≥
1, define the weight zeroPoincar´e series on Γ, P h ( z, s ) := X γ ∈ Γ ∞ \ Γ Im( γz ) s e ( hγz ) , defined initially for Re( s ) sufficiently positive and with meromorphic con-tinuation to all s ∈ C .Let V f,g ( z ) := y k ( f g + T − ( f g )), which is a function in L (Γ \ H ). Byexpanding the Petersson inner product below we get hV f,g , P h ( · , s ) i = Γ ( s + k − π ) s + k − D f,g ( s ; h ) , where we mirror the notation in [HHR13] and define D f,g ( s ; h ) := X n ≥ a ( n ) b ( n − h ) + a ( n − h ) b ( n ) n s + k − , again for Re( s ) sufficiently positive. Dividing by h w and summing over h ≥ Z ( s, w, f × g ), Z ( s, w, f × g ) := X n,h ≥ D f,g ( s ; h ) h w = (4 π ) s + k − Γ( s + k − X h ≥ hV f,g , P h i h w , (4.1)when both Re( s ) and Re( w ) are sufficiently positive.We will obtain a meromorphic continuation of Z ( s, w, f × g ) by using thespectral expansion of the Poincar´e series and substituting it into (4.1). Let { µ j } be an orthonormal basis of Maass eigenforms with associated types + it j for L (Γ \H ) as in Section 2.3, each with Fourier expansion µ j ( z ) = X n =0 ρ j ( n ) y K it j (2 π | n | y ) e πinx . Then the inner product of µ j against the Poincar´e series gives h P h ( · , s ) , µ j i = ρ j ( h ) √ π (4 πh ) s − Γ( s − + it j )Γ( s − − it j )Γ( s ) . (4.2) Remark 4.1.
In the computation of this inner product and the inner prod-uct of the Eisenstein series against the Poincar´e series, we use formula [JZ07, § E ( z, w ) be the Eisenstein series on SL ( Z ) as in (2.3). Then E ( z, w )has Fourier expansion (as in [Gol06, Chapter 3]) given by E ( z, w ) = y w + φ ( w ) y − w (4.3)+ 2 π w √ y Γ( w ) ζ (2 w ) X m =0 | m | w − σ − w ( | m | ) K w − (2 π | m | y ) e πimx , where φ ( w ) = √ π Γ( w − ) ζ (2 w − w ) ζ (2 w ) . The inner product of the Poincar´e series ( h ≥
1) against the Eisenstein series E ( z, w ) is given by h P h ( · , s ) , E ( · , w ) i = 2 π w + h w − σ − w ( h ) ζ (2 w )(4 πh ) s − Γ( s + w − s − w )Γ( w )Γ( s ) , (4.4)provided that Re s > + | Re w − | . For t real, w = + it , and Re s > ,(4.4) specializes to h P h ( · , s ) , E ( · , + it ) i = 2 √ πσ it ( h )Γ( s )(4 πh ) s − Γ( s − + it )Γ( s − − it ) h it ζ ∗ (1 − it ) , (4.5)in which ζ ∗ (2 s ) := π − s Γ( s ) ζ (2 s ) denotes the completed zeta function.The spectral expansion of the Poincar´e series is given by P h ( z, s ) = X j h P h ( · , s ) , µ j i µ j ( z )+ 14 π Z ∞−∞ h P h ( · , s ) , E ( · , + it ) i E ( z, + it ) dt. (4.6)We shall refer to the above sum and integral as the discrete and continuousspectrum, respectively. After substituting (4.2) into the discrete part of(4.6), the discrete spectrum takes the form √ π (4 πh ) s − Γ( s ) X j ρ j ( h )Γ( s − + it j )Γ( s − − it j ) µ j ( z )and is analytic in s in the right half-plane Re s > + θ , where θ =sup j { Im( t j ) } ≤ is the best known progress toward Selberg’s Eigenvalue UMS OF FOURIER COEFFICIENTS 11
Conjecture. After inserting (4.5), the continuous spectrum takes the form √ π π (4 πh ) s − Z ∞−∞ σ it ( h ) h it Γ( s − + it )Γ( s − − it ) ζ ∗ (1 − it )Γ( s ) E ( z, + it ) dt, which has its right-most poles in s when Re s = .Substituting this spectral expansion into (4.1) and executing the sum over h ≥ Proposition 4.2.
For f, g weight k forms on SL ( Z ) , the shifted convolutionsum Z ( s, w, f × g ) can be expressed as Z ( s, w, f × g ) := ∞ X m =1 a ( m ) b ( m − h ) + a ( m − h ) b ( m ) m s + k − h w = (4 π ) k X j ρ j (1) G ( s, it j ) L ( s + w − , µ j ) hV f,g , µ j i (4.7)+ (4 π ) k πi Z (0) G ( s, z ) Z ( s, w, z ) hV f,g , E ( · , − z ) i dz, (4.8) when Re( s + w ) > , where G ( s, z ) and Z ( s, w, z ) are the collected Γ and ζ factors of the discrete and continuous spectra, G ( s, z ) = Γ( s − + z )Γ( s − − z )Γ( s )Γ( s + k − Z ( s, w, z ) = ζ ( s + w − + z ) ζ ( s + w − − z ) ζ ∗ (1 + 2 z ) . Remark 4.3.
Consider Stirling’s approximation, that for x, y ∈ R ,Γ( x + iy ) ∼ (1 + | y | ) x − e − π | y | as y → ±∞ with x bounded. Thus, we have that for vertical strips in s and z , G ( s, z ) ∼ P ( s, z ) e − π (2 max( | s | , | z | ) − | s | ) , where P ( s, z ) has at most polynomial growth in s and z . By Watson’s tripleproduct formula in the integral-weight case, given in Theorem 3 of [Wat08],and Kıral’s bound should we require the half-integral weight case, given inProposition 13 of [Kır15], we know that ρ j (1) h f g Im( · ) k , µ j i and ρ j (1) h T − ( f g ) Im( · ) k , µ j i has at most polynomial growth in | t j | . The same can be said about hV f,g , E ( · , + z ) i /ζ ∗ (1 + 2 z ) , albeit through more direct computation. From this it is clear that (4.7) and(4.8) converge uniformly on vertical strips in t j and have at most polynomialgrowth in s . Meromorphic Continuation.
In this section, we seek to understandthe meromorphic continuation and polar behaviour of Z ( s, , f × g ). Thisnaturally breaks down into two parts: the contribution from the discretespectrum and the contribution from the continuous spectrum.4.2.1. The Discrete Spectrum.
Examination of line (4.7), the contributionfrom the discrete spectrum, reveals that the poles come only from G ( s, it j ).There are apparent poles when s = ± it j − n for n ∈ Z ≥ . Interestingly,the first set of apparent poles are at s = ± it j do not actually occur. Lemma 4.4.
For even Maass forms µ j , we have L ( ± it j , µ j ) = 0 .Proof. The completed L -function associated to a Maass form µ j is given byΛ j ( s ) = π − s Γ (cid:16) s + ǫ + it j (cid:17) Γ (cid:16) s + ǫ − it j (cid:17) L ( s, µ j ) = ( − ǫ Λ j (1 − s ) , (4.9)as in [Gol06, Sec 3.13], where ǫ = 0 if the Maass form µ j is even and 1 if it isodd. As the completed L -function is entire, L ( ± it j , µ j ) is a trivial zero. (cid:3) Similarly, L ( − n ± it j , µ j ) , n ∈ Z ≥ are trivial zeroes for even Maass forms. Lemma 4.5.
Suppose f and g are weight k cusp forms, as above. For oddMaass forms µ j , we have hV f,g , µ j i = 0 .Proof. Recall that V f,g := y k ( f g + T − ( f g )), so clearly T − V f,g = V f,g . Since T − is a self-adjoint operator with respect to the Petersson inner productwe have that hV f,g , µ j i = h T − V f,g , µ j i = hV f,g , T − µ j i = −hV f,g , µ j i . Thus hV f,g , µ j i = 0. (cid:3) Lemmas 4.5 guarantees that the only Maass forms appearing in line (4.7)are even. The first set of apparent poles from even Maass forms appearat s = ± it j and occur as simple poles of the gamma functions in thenumerator of G ( s, t j ). They come multiplied by the value of L ( it j , µ j ),which by Lemma 4.4 is zero.So, in summary, D ( s, S f × S g ) has no poles at s = ± it j . The next setof apparent poles are at s = − ± it j , appearing at the next set of simplepoles of the gamma functions in the numerator. Unlike the previous poles,these do not occur at trivial zeroes of the L -function. So we have poles ofthe discrete spectrum at s = − ± it j .4.2.2. The Continuous Spectrum.
Let us now examine line (4.8), the contri-bution from the continuous spectrum. This is substantially more involvedthan the discrete spectrum and exhibits remarkable cancellation.The rightmost pole seems to occur from the pair of zeta functions in thenumerator, occurring when s + w − ± z = 1. We must disentangle s and w from z in order to understand these poles.Line (4.8) is analytic for Re( s + w ) > . For s with Re s ∈ ( − Re w, − Re w + ǫ ) for some very small ǫ , we want to shift the contour of integration, UMS OF FOURIER COEFFICIENTS 13 avoiding poles coming from the ζ ∗ (1 − z ) appearing in the denominator ofthe expansion of E ( · , + z ). So we shift the z -contour to the right whilestaying within the zero-free region of ζ . By an abuse of notation, we denotethis shift here by Re z = ǫ and let ǫ in this context actually refer to thereal value of the z -contour at the relevant imaginary value. This argumentcan be made completely rigourous, cf. [HHR13, p. 481-483]. By the residuetheorem,(4 π ) k πi Z (0) G ( s, z ) Z ( s, w, z ) hV f,g , E i dz (4.10)= (4 π ) k πi Z ( ǫ ) G ZhV f,g , E i dz − (4 π ) k z = s + w − G ZhV f,g , E i , where the above residue is found to be − ζ (2 s + 2 w − s + w − − w ) ζ ∗ (2 s + 2 w − s )Γ( s + k − hV f,g , E ( · , − s − w ) i . (4.11)The residue is analytic in s for Re s ∈ (1 − Re w, − Re w + ǫ ), and has aneasily understood meromorphic continuation to the whole plane. Notice alsothat the shifted contour integral has no poles in s for Re s ∈ ( − Re w − ǫ, − Re w + ǫ ), so we have found an analytic (not meromorphic!) continuation in s of Line (4.8) past the first apparent pole along Re s = − Re w .For s with Re s ∈ ( − Re w − ǫ, − Re w ), we shift the contour of in-tegration back to Re z = 0. Since this passes a pole, we pick up a residue.But notice that this is the residue at the other pole,(4 π ) k πi Z ( ǫ ) G ( s, w, z ) Z ( s, w, z ) hV f,g , E i dz (4.12)= (4 π ) k πi Z (0) G ZhV f,g , E i dz + (4 π ) k z = − s − w G ZhV f,g , E i . By using the functional equations of the Eisenstein series and zeta functions,one can check thatRes z = − s − w G ZhV f,g , E i = − Res z = s + w − G ZhV f,g , E i . So (4.8), originally defined for Re s > − Re w , has meromorphic continua-tion for − Re w < Re s < − Re w given by(4 π ) k πi Z (0) G ZhV f,g , E i dz + (4 π ) k Res z = − s − w G ZhV f,g , E i . (4.13)A very similar argument works to extend the meromorphic continuationin s of the contour integral past the next apparent pole at Re s = , leading to a meromorphic continuation in the region − < Re s < given by(4 π ) k πi Z (0) G ( s, w, z ) Z ( s, w, z ) hV f,g , E ( · , − z ) i dz (4.14)+ (4 π ) k Res z = − s − w G ( s, w, z ) Z ( s, w, z ) hV f,g , E ( · , − z ) i (4.15)+ (4 π ) k Res z = − s G ( s, w, z ) Z ( s, w, z ) hV f,g , E ( · , − z ) i . (4.16)We iterate this argument, as in Section 4 of [HHR13, p. 481-483]. Some-what more specifically, when Re( s ) approaches a negative half-integer, − n ,we can shift the line of integration for z right past the pole due to G ( s, z )at z = s − + n , move s left past the line Re( s ) = − n and then shift theline of integration for z left, back to zero and over the pole at z = − s − n .This gives meromorphic continuation of (4.8) arbitrarily far to the left, ac-cumulating a pair of residual terms at each half-integer line as is the case in(4.16).We now specialize to w = 0. The rightmost pole of (4.8) occurs in thefirst residual term appearing in (4.15) from the meromorphic continuation.The pole occurs at s = 1 from the Eisenstein series and has residueRes s =1 Res z = s − (4 π ) k G ( s, , z ) Z ( s, , z ) hV f,g , E ( · , − z ) i == Res s =1 (4 π ) k ζ (2 s − s − ζ ∗ (2 s − s )Γ( s + k − hV f,g , E ( · , − s ) i , (4.17)which can be interpreted as (see Section 2.1) − (4 π ) k Γ( k ) 3 π h f Im( · ) k , g i = − Res s =1 L ( s, f × g ) ζ (2) . (4.18)The next pole of (4.8) also occurs in the first residual term appearingin (4.15), occurring at s = from the gamma function in the numera-tor of G ( s, , z ). Otherwise, the continuous spectrum (4.8) is analytic forRe s ≥ . Combining this continuation with the continuation of the discretespectrum (4.7), we get the following lemma. Lemma 4.6.
Maintaining the notation from Proposition 4.2, Z ( s, , f × g ) has meromorphic continuation in s to Re s ≥ with poles at most at s = 1 and s = . The rightmost pole is at s = 1 and has residue − (4 π ) k Γ( k ) 3 π h f Im( · ) k , g i = − Res s =1 L ( s, f × g ) ζ (2) . (4.19) UMS OF FOURIER COEFFICIENTS 15
Returning to the meromorphic continuation of (4.8) given just above, weevaluate the second residual term (4.16), which only appears for Re s < ,Res z = − s (4 π ) k G ( s, , z ) Z ( s, , z ) hV f,g , E ( · , − z ) i (4.20)= (4 π ) k ζ (0) ζ (2 s − s − ζ ∗ (2 − s )Γ( s )Γ( s + k − hV f,g , E ( · , s ) i . (4.21)By using the gamma duplication formulaΓ(2 s − s ) = Γ( s − ) 2 s − √ π and functional equations for ζ ( s ) and E ( z, s ), we can rewrite (4.20) as − L ( s, f × g ) ζ (2 s ) . (4.22)Thus this second residual term has poles at zeroes of ζ (2 s ).More generally, a residual term( − j (4 π ) k Γ( j + 1) ζ ( − j ) ζ (2 s + j − s + j − ζ ∗ (2 − s − j )Γ( s )Γ( s + k − hV f,g , E ( · , s + j ) i (4.23)is introduced for Re s < − j in the continuation past the apparent polarline Re s = − j for each integer j ≥
0. We recognize (4.21) as the j = 0case of (4.23). We note that the first residual term (4.15) is distinguished incoming from a pole from the zeta function, while all further residual termshave the same form as (4.23) and come from poles from gamma functions. Asin (4.22), the Eisenstein series appearing in the j th residual (4.23) introducespoles at s = ρ − j for each nontrivial zero ρ of ζ ( s ).4.3. Analytic Behaviour of W ( s ; f, g ) . Recall that W ( s ; f, g ) = L ( s, f × g ) ζ (2 s ) + Z ( s, , f × g ) . Then Lemma 4.6 shows that the leading pole of L ( s,f × g ) ζ (2 s ) at s = 1 cancelsperfectly with the leading pole of Z ( s, , f × g ). So W ( s ; f, g ) is analyticfor Re s > and has a pole at s = . Further, since the meromorphiccontinuation of Z ( s, , f × g ) has (4.22) as a residual for Re s < , we see thatthe Rankin-Selberg L -function L ( s,f × g ) ζ (2 s ) term completely cancels when Re s < . Then W ( s ; f, g ) has a meromorphic continuation to C , and for Re s > − the only possible poles are s = , coming from the first residual term (4.15)of the continuous spectrum (4.8) and those at s = − ± it j , coming fromthe exceptional eigenvalues of the discrete spectrum (4.7). (There are noexceptional eigenvalues for Γ = SL ( Z )). Let us evaluate the residue of Z ( s, , f × g ) at the pole s = . For ease,we write the first residual term as a residue at z = − s ,(4 π ) k ζ (2 s − s − ζ ∗ (2 s − s )Γ( s + k − hV f,g , E ( · , − s ) i . (4.24)By applying the gamma duplication formula, expanding the completed zetafunction in the denominator and cancelling similar terms from the numeratorand denominator, this becomes(4 π ) s + k − √ π Γ( s − )Γ( s )Γ( s + k − hV f,g , E ( · , − s ) i . (4.25)There is a pole at s = coming from the gamma function in the numerator.The residue at this pole is given by12 √ π Γ( ) (4 π ) k − Γ( k − ) hV f,g , E ( · , ) i . (4.26)We rewrite this as a special value of the Rankin-Selberg L -function12 π ( k − )4 π (4 π ) k + Γ( k + ) hV f,g , E ( · , ) i = ( k − )4 π L ( , f × g ) ζ (3) . (4.27)This allows us to conclude the following theorem. Theorem 4.7.
Let f, g be two holomorphic cusp forms on SL ( Z ) . Main-taining the same notation as above, the function W ( s ; f, g ) has a meromor-phic continuation to C given by (2.2) and Proposition 4.2 with potentialpoles at s with Re s ≤ and s ∈ Z ∪ ( Z + ) ∪ S ∪ Z , where Z denotes theset of shifted zeta-zeroes {− ρ − n : n ∈ Z ≥ } , and S denotes the set ofshifted discrete types {− ± it j − n : n ∈ Z ≥ } .The leading pole is at s = and Res s = W ( s ; f, g ) = ( k − )4 π L ( , f × g ) ζ (3) . (4.28)With this theorem and the decomposition from Proposition 3.1, we havethe meromorphic continuation of the Dirichlet series D ( s, S f × S f ). Remark 4.8.
Very similar work gives the meromorphic continuation for D ( s, S f × S g ), mainly replacing g with T − g in the above formulation. Thisdistinction only matters at higher levels when f and g have nontrivial neben-typus, and the spectral expansion is modified accordingly.5. Second Moment of Sums of Fourier Coefficients
Now we will use the results of the previous section to prove Theorem 1.1for cusp forms on SL ( Z ) and suggest how the argument generalizes. First,we state a simple corollary of Theorem 4.7. UMS OF FOURIER COEFFICIENTS 17
Corollary 5.1.
Let θ = max j Im( t j ) ≤ denote the progress towardsSelberg’s Eigenvalue Conjecture for the given congruence subgroup Γ . Thefunction W ( s ; f, g ) = L ( s, f × g ) ζ (2 s ) + Z ( s, , f × g ) , appearing in Proposition 3.1, is analytic for Re s > − + θ except for asimple pole at s = . We now consider a smooth cutoff integral of D ( s, S f × S g ). Using thewell-known integral transform,12 πi Z ( σ ) D ( s, S f × S g ) X s Γ( s ) ds = X n ≥ S f ( n ) S g ( n ) n k − e − n/X , (5.1)for σ large enough to be in the domain of absolute convergence of D ( s, S f × S g ), say σ = 4. To understand the right hand side of (5.1), we decompose theleft hand side as in Proposition 3.1. We thus investigate the two integrals,12 πi Z (4) W ( s ; f, g ) X s Γ( s ) ds (5.2)and 1(2 πi ) Z (4) Z (2) W ( s − z ; f, g ) ζ ( z ) Γ( z )Γ( s − z + k − s + k − dz X s Γ( s ) ds. (5.3) Lemma 5.2.
Fix an ǫ > . Then the integral (5.2) is ( k − )4 π L ( , f × g ) ζ (3) Γ( ) X / + O ǫ ( X ǫ ) , (5.4) where O ǫ ( · ) indicates that the implicit constant depends on ǫ .Proof. Shifting the line of integration to Re s = ǫ passes the pole at s = mwith residue as given by Theorem 4.7. To bound the shifted integral, weobserve that W ( s ; f, g ) has at most polynomial growth in vertical stripswhile Γ( s ) has exponential decay. (cid:3) Lemma 5.3.
Fix an ǫ > . Then the integral (5.3) is π L ( , f × g ) ζ (3) Γ( ) X / + O ǫ ( X + θ + ǫ ) . (5.5) Proof.
We first shift the z line of integration to ǫ , passing a pole at z = 1from ζ ( z ) with residue12 πi Z (4) W ( s − f, g ) 1 s + k − X s Γ( s ) ds. (5.6)The remaining analysis of this integral is almost identical to the analysisof (5.2). We shift the line of integration to Re s = + θ + ǫ , passing overthe pole at s = . The integrand has exponential decay in vertical strips,so the s -shifted integral is O ǫ ( X + θ + ǫ ). All that remains is the shifted double integral1(2 πi ) Z (4) Z ( ǫ ) W ( s − z ; f, g ) ζ ( z ) Γ( z )Γ( s − z + k − s + k − dz X s Γ( s ) ds. (5.7)We shift the line of s integration to Re s = + 2 ǫ without encountering anypoles. Again we have exponential decay in vertical strips in both s and z , sowe can conclude that this integral is also O ǫ ( X
12 +2 ǫ ). Putting these togethergives the lemma. (cid:3) By combining Lemmas 5.2 and 5.3, we have proved Theorem 1.1 for D ( s, S f × S g ), as stated in the introduction, for level one. Theorem. If f and g are holomorphic cusp forms for SL ( Z ) , then for any ǫ > , X X n ≥ S f ( n ) S g ( n ) n k − e − n/X = CX + O f,g,ǫ ( X − + θ + ǫ ) where C = Γ( )4 π L ( , f × g ) ζ (3) = Γ( )4 π X n ≥ a ( n ) b ( n ) n k + . Remark 5.4.
The main term in the above theorem, CX , mostly agreeswith the main term for the sharp cut-off in [HI89] as stated in (1.2) and (1.4)where f = g , though there are a few notable differences. Here we havenormalized the Dirichlet series by dividing each | S f ( n ) | term by n k − , sothe power of X is rather than k − . Furthermore, as we are performinga smooth cut-off, we have an extra factor of Γ( ) owing to the Γ( s ) in theinverse Mellin transform in (5.1). It is not hard to check that performing asharp cut-off on the non-normalized Dirichlet series by integrating against1 /s instead of Γ( s ) gives the exact same formula for the main term in (1.4). Remark 5.5.
When proving the analogue of this theorem for D ( s, S f × S g ),there are only significant differences when f and g have nontrivial associatednon-real nebentypus. As noted before case, the primary differences comefrom slightly more complicated Eisenstein series, L -functions associated tothe Eisenstein series, and functional equations for these L -functions. Weleave this out of the statement of the above theorem since for SL ( Z ) allautomorphic forms are complex linear combinations of eigenforms with realcoefficients, so the distinction between S f ( n ) S g ( n ) and S f ( n ) S g ( n ) is trivial. Remark 5.6.
It is natural to try to shift the lines of integration furtherleft, but this does not give much improvement. In Section 4, we see that Z ( s, , f × g ) has a line of poles when Re s = − + θ , indicating that theexponent of the error term in Theorem 1.1 cannot be lowered.If one could prove a sharp cutoff instead of a smoothed sum of the aboveshape, one could prove the Classical Conjecture. UMS OF FOURIER COEFFICIENTS 19 Cancellation within W ( s ; f, g ) for arbitrary level While the techniques and methodology of Section 4 should work forgeneral level, it is not immediately clear that the continuous spectrum of Z ( s, , f × g ) will always perfectly cancel the leading pole and potentialpoles from zeta zeroes of L ( s,f × g ) ζ (2 s ) . In this section, we apply results of Chan-drasekharan and Narasimhan ([CN62] and [CN64]) that show that signifi-cant cancellation always occurs in the Mellin integrals, shedding new light oncancellation of terms between the diagonal and off-diagonal parts of shiftedconvolution sums and on the behavior of certain sums of Kloosterman zetafunctions.We then explicitly show that further cancellation holds for integral weightmodular forms on Γ ( N ) when N is square-free, and completely generalizeTheorem 1.1 to squarefree level.6.1. Leading Cancellation.
Suppose f ( z ) = P a ( n ) e ( nz ) is a cusp formon the congruence subgroup Γ, of weight k ∈ Z ∪ ( Z + ) with k >
2. Wealso allow the possibility for f to have nebentypus χ if Γ = Γ ( N ) for some N ∈ N . Theorem 1 of [CN64] gives that1 X X n ≤ X | S f ( n ) | n k − = CX + O (log X ) . (6.1)Let us compare the result of Chandrasekharan and Narasimhan to themethodology of this paper. Performing the decomposition of D ( s, S f × S f )from Proposition 3.1 still leads us to study Z ( s, , f × f ) and W ( s ; f, f ).The Rankin-Selberg convolution L ( s, f × f ) /ζ (2 s ) has poles at s = 1 andat zeroes of ζ (2 s ) in (0 , ). The contributions from these poles must cancelwith those from the poles of Z ( s, , f × f ) in the inverse Mellin transformfrom which Theorem 1.1 is derived, as otherwise the machinery of Sections 4and 5 contradict (6.1). In particular, the leading contribution of the diagonalterm cancels perfectly with a leading contribution from the off-diagonal,Res s =1 X n ≥ | a ( n ) | n s + k − = − Res s =1 X n,h ≥ a ( n ) a ( n − h ) + a ( n − h ) a ( n ) n s + k − . We investigate this cancellation further by sketching the arguments of Sec-tions 4 and 5 in greater generality.The spectral decomposition corresponding to Proposition 4.2 is more com-plicated since we must now use the Selberg Poincar´e series on Γ, P h ( z, s ) := X γ ∈ Γ ∞ \ Γ ( N ) Im( γz ) s e ( hγ · z ) . (6.2)The spectral decomposition of P h will involve Eisenstein series associated toeach cusp, a , of Γ, which each have an expansion, E a ( z, w ) = δ a y w + ϕ a (0 , w ) y − w + X m =0 ϕ a ( m, w ) W w ( | m | z ) , (6.3) where δ a = 1 if a = ∞ and is 0 otherwise, where ϕ a (0 , w ) = √ π Γ( w − )Γ( w ) X c c − w S a (0 , c ) , and ϕ a ( m, w ) = π w Γ( w ) | m | w − X c c − w S a (0 , n ; c )when m = 0, are generalized Whittaker-Fourier coefficients, W w ( z ) = 2 √ yK w − (2 πy ) e ( x )is a Whittaker function, K ν ( z ) is a K -Bessel function, and S a ( m, n ; c ) = X ( a · c d ) ∈ Γ ∞ \ σ − α Γ / Γ ∞ e (cid:18) m dc + n ac (cid:19) is a Kloosterman sum associated to double cosets of Γ withΓ ∞ = (cid:28)(cid:18) n (cid:19) : n ∈ Z (cid:29) ⊂ SL ( Z ) . This expansion is given in Theorem 3.4 of [Iwa02].Letting µ j be an orthonormal basis of the residual and cuspidal spaces, wemay expand P h ( z, s ) by the Spectral Theorem (as presented in Theorem 15.5of [IK04]) to get P h ( z, s ) = X j h P h ( · , s ) , µ j i µ j ( z ) (6.4)+ X a π Z R h P h ( · , s ) , E a ( · , + it ) i E a ( z, + it ) dt. (6.5)This is more complicated than the SL ( Z ) spectral expansion in (4.6) fortwo major reasons: we are summing over cusps and the Kloosterman sumswithin the Eisenstein series are trickier to handle. Continuing as before, wetry to understand the shifted convolution sum Z ( s, w, f × f ) = (4 π ) s + k − Γ( s + k − X h ≥ hV f,f , P h i h w (6.6)by substituting the spectral expansion for P h ( z, s ) and producing a mero-morphic continuation.The analysis of the discrete spectrum is almost exactly the same: it isanalytic for Re s > − + θ , and so the only new facet is understanding thecontinuous spectrum component corresponding to (6.5). As noted above,the continuous spectrum of Z ( s, , f × f ) has leading poles that perfectlycancel the leading pole of L ( s, f × f ) ζ (2 s ) − . Using analogous methods to UMS OF FOURIER COEFFICIENTS 21 those in Section 4, we compute the continuous spectrum of Z ( s, , f × f ) toget X h ≥ (4 π ) s + k − Γ( s + k − X a πi Z ( ) h P h ( · , s ) , E a ( · , t ) ihV f,f , E a ( · , t ) i dt = (4 π ) k Γ( s + k − s ) X a Z ∞−∞ X h,c ≥ S a (0 , h ; c ) h s + it c − it π − it Γ( − it ) ×× Γ( s − + it )Γ( s − − it ) hV f,f , E a ( · , + it ) i dt. (6.7)We’ve placed parentheses around the arithmetic part, including the Kloost-erman sums and factors for completing a zeta function that appears withinthe Kloosterman sums. Remark 6.1.
We note also that (1.2), as in [CN62], suggests that D ( s, S f × S f ) has a pole at s = and that no other poles should contribute additionalterms to the inverse Mellin-transform for Re s >
0. As in Section 4.2,we recognize the pole at s = coming from the continuous spectrum of Z ( s, , f × f ).However, the exact nature of the potential polar behavior for Re s ∈ (0 , )cannot be determined completely by comparison with [CN62]. It is naturalto conjecture that there are no poles of D ( s, S f × S f ) with Re s > s = .We can make similar claims about the cancellation in the case of D ( s, S f × S g ) when f = g . Indeed, if we let h = f + g and h = f + ig , then we havethat | S h ( n ) | = | S f ( n ) | + | S g ( n ) | + 2 Re (cid:16) S f ( n ) S g ( n ) (cid:17) and | S h ( n ) | = | S f ( n ) | + | S g ( n ) | + 2 Im (cid:16) S f ( n ) S g ( n ) (cid:17) . Since D ( s, S h i × S h i ) , D ( s, S f × S f ) , and D ( s, S g × S g ) do not have polesfor Re( s ) > , it follows that the continuations of ∞ X n =1 Re (cid:16) S f ( n ) S g ( n ) (cid:17) n s + k − and ∞ X n =1 Im (cid:16) S f ( n ) S g ( n ) (cid:17) n s + k − are also meromorphic in this region. Thus this is also the case for D ( s ; S f × S g ).We summarize the results of this section with the following theorem. Theorem 6.2.
Let f ( z ) = P a ( n ) e ( nz ) and g ( z ) = P b ( n ) e ( nz ) be weight k > cusp forms on Γ , possibly with nebentypus χ if Γ = Γ ( N ) . Then Res s =1 X n ≥ a ( n ) b ( n ) n s + k − = − Res s =1 X n,h ≥ a ( n ) b ( n − h ) + a ( n − h ) b ( n ) n s + k − , (6.8) or equivalently − Res s =1 L ( s, f × g ) ζ (2 s ) = (6.9)= Res s =1 X a ∞ X h =1 π Z R h P h ( · , s ) , E a ( · , + it ) ihV f,g , E a ( · , + it ) i dt (6.10)= Res s =1 (4 π ) k Γ( s + k − s ) X a Z ∞−∞ X h,c ≥ S a (0 , h ; c ) h s + it c − it π − it Γ( − it ) × Γ( s − + it )Γ( s − − it ) hV f,g , E a ( · , + it ) i dt. (6.11) Remark 6.3.
The result is essentially the same and the above argument islargely unchanged if we replace b ( n ) by b ( n ). The only potential issue is ifthere is an accompanying non-real nebentypus, χ , for f and g . In this case,we change the group we are considering to Γ ( N ), where the nebentypusdoes not affect the proof. Since g and T − g are both weight k holomorphiccusp forms for Γ ( N ), we use that D ( s ; S f × S g ) = D ( s ; S f × S T − g ) and weget that D ( s ; S f × S g ) is also meromorphic for Re( s ) > .6.2. Complete Cancellation for Square-free Level.
Now we considerthe case where the level N is square-free, where our methodology explicitlydemonstrates the same complete, remarkable cancellation as in level 1. LetΓ = Γ ( N ) where N is square-free. A complete set of cusp representativesis given by { v : v | N } . As in [DI82], we can derive more explicit formulasfor ϕ a ( m, w ) when a = v , ϕ a (0 , w ) = ζ ∗ (2 s − ζ ∗ (2 s ) (cid:18) vN (cid:19) s ϕ ( v ) Y p | N (cid:18) − p s (cid:19) − Y p | Nv (cid:18) − p s − (cid:19) and ϕ a ( m, w ) = 2 π s | m | s − Γ( s ) (cid:18) vN (cid:19) s σ ( N )1 − s ( m ) ζ ( N ) (2 s ) Y p | vp α k m ( pσ − s (cid:0) p α − ) − σ − s ( p α ) (cid:1) , where σ ( c ) v ( n ) = X d | n ( d,c )=1 d v and ζ ( N ) (2 s ) = ζ (2 s ) Y p | N (1 − p − s ) . UMS OF FOURIER COEFFICIENTS 23
From Hejhal [Hej83], we know that the Eisenstein series satisfy the functionalequation E ∞ ( z, s ) = ζ ∗ (2 − s ) ζ ∗ (2 s ) Y p | N − p s X v | N Y p | v (1 − p ) Y p | Nv ( p − s − p s ) E v ( z, − s ) . (6.12)Using these coefficients, we can explicitly describe write the continuouspart of the spectrum of Z ( s, w, f × g ) in (6.11) as(4 π ) k Γ( s + k −
1) 14 πi Z (0) X v | N h V f,g , E v ( ∗ , − z ) i Γ( s − + z )Γ( s − − z )Γ( s )( vN ) + z × ζ ( s + w − − z ) ζ ( Nv ) ( s + w − + z ) π − ( + z ) Γ( + z ) ζ ( N ) (1 + 2 z ) Y p | v ( p + z − s − w − dz. (6.13)For Re( s + w ) close to , we obtain residual terms from the poles at z = ± ( s + w − ) through the same procedure for moving lines of integrationfrom Section 4. For z = s + w − , there is a pole only when the productover primes dividing v , as otherwise the simple pole from the zeta functionis cancelled by the zero of the product. Therefore, a pole occurs only when v = 1. In that case, the z residue is:(4 π ) k s + k − h V f,g , E ( ∗ , − s − w ) i× Γ(2 s + w − − w )Γ( s + w − s ) π s + w − N s + w − . (6.14)Setting w = 0 and then taking residue at s = 1, we get − (4 π ) k k ) Res s =1 h V f,g , E ( ∗ , s ) i = −
12 Res s =1 X n ≥ a ( n ) b ( n ) n s + k − . When z = − s − w , there is a pole from the zeta function in the numeratorwith residue(4 π ) k s + k − X v | N h V f,g , E v ( ∗ , s + w − i Γ(1 − w )Γ(2 s + w − s )( vN ) − s − w × ζ (2 s + 2 w − Q p | Nv (1 − p ) Q p | v ( p − s − w − π − (2 − s − w ) Γ(2 − s − w ) ζ ( N ) (4 − s − w ) . (6.15)We set w = 0. Using the functional equation from Hejhal [Hej83], which forsquare-free N gives E ∞ ( z, s ) = ζ ∗ (2 − s ) ζ ∗ (2 s ) Y p | N − p s X v | N Y p | v (1 − p ) Y p | Nv ( p − s − p s ) E v ( z, − s ) and E ( z, s ) = ζ ∗ (2 − s ) ζ ∗ (2 s ) Y p | N − p s X v | N Y p | Nv (1 − p ) Y p | v ( p − s − p s ) E v ( z, − s ) . Notice that this is the same as the residue at z = s + w − , so we can explic-itly verify the cancellation of poles of Z ( s, , f × g ) at s = 1 as guaranteedby Theorem 6.2.From the continuous spectrum (6.13), there are residual terms comingfrom z = ± ( s + w − ). In Section 4, setting w = 0 and examining the corre-sponding residual term in the level 1 case led to a term perfectly cancellingthe diagonal contribution L ( s, f × f ) /ζ (2 s ).We now investigate this residual term and cancellation in the square-freelevel case. Recall that we now have w = 0. The residual term from z = − s ,occurring when z = − s in the zeta function ζ ( Nv ) ( s − + z ) in the numeratorof (6.13), vanishes except when v = N since ζ ( Nv ) (0) = 0 if Nv >
1. Theresidual term when v = N is − (4 π ) k s + k − h V f,g , E ∞ ( ∗ , s ) i π − s N s − Γ(2 s − ζ (2 s − s )Γ(1 − s ) ζ ( N ) (2 − s ) Y p | N ( p − s − . Using duplication formula and functional equation for the completed zetafunction, this simplifies to − X n ≥ a ( n ) b ( n ) n s + k − . (6.16)(Note that this is very similar to the analysis of (4.21)).The residual term from z = s − coming from the other zeta function inthe numerator of (6.13) is(4 π ) k s + k − X v | N h V f,g , E v ( ∗ , − s ) i Γ(2 s − s )( vN ) s ζ (0) ζ ( Nv ) (2 s − π − s Γ( s ) ζ ( N ) (2 s ) Y p | v ( p − . Using the functional equation (6.12) of the Eisenstein series and the relation ζ (0) ζ ( Nv ) (2 s − π − s ( vN ) s ζ ( N ) (2 s ) Y p | v ( p − ζ (2 s − vN ) s ζ ∗ (2 s ) Q p | v ( p − Q p | Nv (1 − p − s ) Q p | N (1 − p − s )= π s − Γ( s − ) ζ ∗ (2 − s ) ζ ∗ (2 s ) Q p | v ( p − Q p | Nv ( p s − p − s ) Q p | N ( p s − , UMS OF FOURIER COEFFICIENTS 25 we have(4 π ) k s + k − X v | N h V f,g , E v ( ∗ , − s ) i Γ(2 s − s )( vN ) s ζ (0) ζ ( Nv ) (2 s − π − s Γ( s ) ζ ( N ) (2 s ) Y p | v ( p − − (4 π ) k s + k − π s − − s h V f,g , E ∞ ( ∗ , s ) i = − X n ≥ a ( n ) b ( n ) n s + k − . This is exactly the same as (6.16), and together these add together toperfectly cancel L ( s, f × g ) ζ (2 s ) − . Thus we see these residual terms in Z ( s, , f × g ) perfectly cancel the Rankin-Selberg diagonal for Re s ≤ .Finally, as in Section 4, we conclude that D ( s, S f × S g ) is analytic forRe( s ) > − + θ , except for a simple pole at s = coming from (6.14)and (6.15). Here, θ = max j Im( t j ) ≤ denotes the progress towards Sel-berg’s Eigenvalue Conjecture.In other words, the meromorphic properties of D ( s, S f × S g ) for square-free level are very similar to the meromorphic properties on level 1. It isnow clear that performing the exact same Mellin transform analysis as inSection 5 gives the following generalization of Theorem 1.1. Theorem 6.4.
Suppose that f and g are holomorphic cusp forms of integralweight k on Γ ( N ) , N square-free. Then for any ǫ > , X X n ≥ S f ( n ) S g ( n ) n k − e − n/X = CX + O f,g,ǫ ( X − + θ + ǫ ) (6.17) where C is is a calculable residue depending only on f and g , and θ =max j Im( t j ) denotes the progress towards Selberg’s Eigenvalue Conjecturefor Γ ( N ) . Acknowledgements
We would like to thank Jeff Hoffstein at Brown University, who introducedus to this problem by asking a question at a talk given by Winfried Kohnenat Jeff’s 61st birthday conference, and who has besides mentored us all.
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