Short-Interval Averages of Sums of Fourier 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 SHORT-INTERVAL AVERAGES OF SUMS OF FOURIERCOEFFICIENTS OF CUSP FORMS
THOMAS A. HULSE, CHAN IEONG KUAN, DAVID LOWRY-DUDA,AND ALEXANDER WALKER
Abstract.
Let f be a weight k holomorphic cusp form of level one, andlet S f ( n ) denote the sum of the first n Fourier coefficients of f . In anal-ogy with Dirichlet’s divisor problem, it is conjectured that S f ( X ) ≪ X k − + + ǫ . Understanding and bounding S f ( X ) has been a very ac-tive area of research. The current best bound for individual S f ( X ) is S f ( X ) ≪ X k − + (log X ) − . from Wu [Wu09].Chandrasekharan and Narasimhan [CN62] showed that the ClassicalConjecture for S f ( X ) holds on average over intervals of length X . Ju-tila [Jut87] improved this result to show that the Classical Conjecturefor S f ( X ) holds on average over short intervals of length X + ǫ . Build-ing on the results and analytic information about P | S f ( n ) | n − ( s + k − from our recent work [HKLDW15], we further improve these results toshow that the Classical Conjecture for S f ( X ) holds on average over shortintervals of length X (log X ) . Introduction and Statement of Results
Let f be a holomorphic cusp form of integer weight k ≥ ( Z ), andlet the Fourier expansion of f at the cusp at infinity be given by f ( z ) = X n ≥ a ( n ) e ( nz ) , where e ( z ) := e πiz . We adopt the convention that a ( n ) = 0 for n ≤
0. Let S f ( n ) denote the n th partial sum, S f ( n ) := X m ≤ n a ( m ) . Folowing Deligne’s celebrated bound [Del74], we have a ( m ) ≪ m k − + ǫ ,which gives the trivial bound S f ( n ) ≪ n k − +1+ ǫ . One might assume thatthe signs of each a ( m ) are random, which would produce a “square-root 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. cancellation estimate” S f ( n ) ≪ n k − + + ǫ . However, significantly more can-cellation is known. Deligne [Del74] showed that S f ( n ) ≪ n k − + + ǫ . (1.1)Jutila [Jut87] and Hafner and Ivi´c [HI89] removed the ǫ in the above boundand Rankin [Ran90] showed how to get logarithmic savings. More recently,Wu [Wu09] showed that S f ( n ) ≪ n k − + (log n ) − . and used this result to improve lower bounds on numbers of Hecke eigenval-ues of the same signs.In the other direction, Hafner and Ivi´c [HI89] showed that, for some pos-itive constant D , S f ( n ) = Ω ± n k − + D (log log x ) / (log log log x ) / !! . After some normalization, this result resembles the conjectured -exponentin the Gauss circle and Dirichlet divisor problems, and similarly it is naturalto investigate the best exponent, α , such that S f ( n ) ≪ n k − + α + ǫ . As withthe circle and divisor problems, it is speculated that S f ( n ) ≪ n k − + + ǫ (1.2)and we refer to this hypothesis as the “Classical Conjecture,” after Hafnerand Ivi´c.Chandrasekharan and Narasimhan [CN62, CN64] proved that the Clas-sical Conjecture is true on average in long intervals by showing that for X ≫
1, 1 X X n ≤ X | S f ( n ) | = CX k − + O (cid:16) X k − ǫ (cid:17) , (1.3)with an explicit constant C , which gives (1.2) on average via the Cauchy-Schwarz inequality.Jutila [Jut87] improved (1.3) by showing that the Classical Conjecture istrue on average in intervals of length X + ǫ around X . Very recently, Ernvall-Hyt¨onen [EH15] extended Jutila’s work and investigated average orders ofshort sums of Fourier coefficients.Short-interval average results can be used to derive bounds for individual S f ( n ). In particular, the Classical Conjecture would follow from a proof ofthe Classical Conjecture on average in intervals of length X + ǫ around X .Here, we show that the Classical Conjecture holds on average in intervalsof length less than X + ǫ . In particular, we prove the following theorem(we actually prove slightly more, see Remark 2.7), which shows that theClassical Conjecture is true on average within intervals around X of length X (log X ) . HORT-INTERVAL AVERAGES OF SUMS OF FOURIER COEFFICIENTS OF CUSP FORMS3
Theorem 1.1.
Let f be a full integer weight k ≥ cusp form on SL ( Z ) .Then X (log X ) X | n − X | In this section, we decompose D ( s, S f × S f ) into pieces we can under-stand and introduce the concentrating integral transform that leads to ourmain theorem. The majority of this paper is dedicated towards boundingcontributions from each part of this decomposition. We outline the mainargument of the paper, leaving proofs to later sections.From Section 3 of [HKLDW15], we have the following decomposition of D ( s, S f × S f ). Proposition 2.1. Let f ( z ) = X n ≥ a ( n ) e ( nz ) be a weight k cusp form. Then D ( s, S f × S f ) := X n ≥ | S f ( n ) | n s + k − = W ( s ; f, f ) + W ( s − f, f ) s + k − 2+ 12 πi Z ( σ z ) W ( s − z ; f, f ) ζ ( z ) Γ( z )Γ( s + k − − z )Γ( s + k − dz, where Re σ z ∈ (0 , , Re s is sufficiently positive, and W ( s ; f, f ) := L ( s, f × f ) ζ (2 s ) + Z ( s, , f × f ) , HULSE, KUAN, LOWRY-DUDA, AND WALKER in which Z ( s, w, f × f ) is the symmetrized shifted convolution sum Z ( s, w, f × f ) := X n,h ≥ a ( n ) a ( n − h ) + a ( n − h ) a ( n ) n s + k − h w . Here and throughout, we maintain the notation from [HKLDW15], whichlargely reflects the notation in [HHR16]. The analytic properties of D ( s, S f × S f ) are given by the analytic properties of the Rankin-Selberg convolution L ( s, f × f ) and the shifted convolution sum Z ( s, w, f × f ). We understand L ( s, f × f ) through the well-known relation L ( s, f × f ) = (4 π ) s + k − ζ (2 s )Γ( s + k − D | f | Im( · ) k , E ( · , s ) E , where E ( z, s ) is the real analytic Eisenstein series E ( z, s ) := X γ ∈ Γ ∞ \ SL ( Z ) Im( γz ) s . We understand Z ( s, w, f × f ) through meromorphic continuation of its spec-tral expansion. Let { µ j ( z ) : j ≥ } be an orthonormal basis of weight zeroMaass eigenforms for the residual and cuspidal spaces of SL ( Z ) \ H . Thisbasis consists of the constant function µ ( z ) and infinitely many Maass cuspforms µ j ( z ), with j ≥ 1. To these cusp forms we associate eigenvalues + t j with respect to the hyperbolic Laplacian, and Fourier expansions µ j ( z ) = X n =0 ρ j ( n ) y K it j (2 π | n | y ) e πinx . We may assume as well that each µ j is a simultaneous eigenfunction of theHecke operators, including the T − operator which has action T − µ j ( x + iy ) = µ j ( − x + iy ), as described in [Gol06, Theorem 3.12.6].Following [HKLDW15], let V f,f ( z ) = Im( z ) k ( | f ( z ) | + T − ( | f ( z ) | )) . The following Proposition gives the spectral expansion of Z ( s, w, f × f ) interms of the Eisenstein series and the Maass eigenforms defined above. Proposition 2.2 (From Section 4 of [HKLDW15]) . In the region Re s > and Re ( s + w ) > , the shifted convolution sum Z ( s, w, f × f ) can beexpressed as Z ( s,w, f × f ) =(4 π ) k X j ρ j (1) G ( s, it j ) L ( s + w − , µ j ) hV f,f , µ j i (2.1)+ (4 π ) k πi Z (0) G ( s, z ) Z ( s, w, z ) hV f,f , E ( · , − z ) i dz. (2.2) HORT-INTERVAL AVERAGES OF SUMS OF FOURIER COEFFICIENTS OF CUSP FORMS5 We call (2.1) the discrete spectrum component and (2.2) the continuousspectrum component. Here, G ( s, z ) and Z ( s, w, z ) are the collected gammaand zeta 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 ) , in which ζ ∗ (2 z ) := π − z Γ( z ) ζ (2 z ) is the completed zeta function. The con-volution Z ( s, , f × f ) has a meromorphic continuation to the whole planegiven by the separate meromorphic continuations of its discrete and contin-uous spectra. Specializing to w = 0, the discrete spectrum component has a mero-morphic continuation to the plane given by the analytic continuations of L ( s, µ j ). The continuous spectrum also has a meromorphic continuation,but it is significantly more involved due to interaction between poles in s and the Mellin integral in z . The meromorphic continuation of the contin-uous spectrum component can be written as(4 π ) k πi Z (0) G ( s, z ) Z ( s, , z ) hV f,f , E ( · , − z ) i dz + X ≤ m< − Re s ρ − m ( s ) , (2.3)where ρ ( s ) = (4 π ) k ζ (2 s − s − s )Γ( s + k − ζ ∗ (2 s − hV f,f , E ( · , − s ) i (2.4)denotes the residual term coming from the apparent pole of the zeta func-tions in Z ( s, , z ), and for m ≥ ρ − m ( s ) = ( − m − (4 π ) k ζ (1 − m ) ζ (2 s + m − s + m − m )Γ( s )Γ( s + k − ζ ∗ (4 − s − m ) × (cid:10) V f,f , E ( · , s + m − (cid:11) (2.5)denotes residual terms coming from apparent poles of the Gamma functionsin G ( s, z ). Each residual term ρ − m ( s ) appears only when Re s < − m .For example, the first residual term, ρ ( s ), appears when Re s < , and thenext residual term, ρ ( s ), appears when Re s < . When Re ( s ) = − m ,the line of integration in (2.3) is bent so that it does not pass through polesof the integrand.We use the Mellin transform,12 πi Z (2) exp (cid:18) πs y (cid:19) X s y ds = 12 π exp (cid:18) − y log X π (cid:19) , (2.6)for X ≫ y > 0, to concentrate sums around the coefficients of D ( s, S f × S f ) in an interval. In Section 4 we prove (2.6) and investigate HULSE, KUAN, LOWRY-DUDA, AND WALKER a family of related Mellin transforms. Note that12 πi Z ( σ s ) D ( s, S f × S f ) exp (cid:18) πs y (cid:19) X s y ds (2.7)= 12 π X n ≥ | S f ( n ) | n k − exp (cid:18) − y log ( X/n )4 π (cid:19) , (2.8)when Re σ s is large enough such that D ( s, S f × S f ) converges absolutely.Taking Re σ s > n ∈ [ X − X/y, X + X/y ], the exponentialdamping term in line (2.8) is almost constant. But for n with | n − X | >X/y − ǫ , the damping term contributes exponential decay. So the main con-tribution to the integral in (2.7) concentrates around those coefficients in aninterval around X . Further, by the positivity of the coefficients, X | n − X | See Section 5. (cid:3) The third term, coming from (2.12), the z -integral of W ( s − z ; f, f ), re-quires more attention. We fix δ > y ≪ X − δ , eventually taking δ near . Under this assumption, we are ableto shift our lines of integration for both the z and s integrals in a way thatminimizes our final estimates.To be specific, with y ≪ X − δ , it becomes advantageous to shift σ s tobe very negative and σ z to be very positive even though these shifts pass HORT-INTERVAL AVERAGES OF SUMS OF FOURIER COEFFICIENTS OF CUSP FORMS7 over several poles. The number of residual contributions from these polesdepends on the choice of δ , and the resulting implicit constant might be verylarge in δ . Ultimately, we are able to show that the contribution from (2.12)is no larger than that seen in Lemma 2.3, up to implicit dependence on δ . Lemma 2.4. Z ( σ s ) Z ( ǫ ) W ( s − z ; f, f ) ζ ( z ) B ( s + k − − z, z ) exp (cid:18) πs y (cid:19) X s y dzds ≪ δ,ǫ X + ǫ y where B ( s, z ) is the Beta function B ( s, z ) = Γ( z )Γ( s )Γ( s + z ) . Proof. See Section 7. (cid:3) For the term (2.11), coming from W ( s − f, f ) / ( s + k − W ( s ; f, f ) in two smaller pieces.In Section 6.1 we consider the contribution from L ( s, f × f ) ζ (2 s ) − and thecontinuous spectrum of Z ( s, , f × f ), which we group together to take ad-vantage of the remarkable cancellation of polar terms. In our methodology,it becomes necessary to further assume that y ≪ X − δ with δ > . Withthis extra assumption, it is advantageous to shift the line of s -integrationfar to the left.Recall from (2.3) that the meromorphic continuation of Z ( s, , f × f ) gainsadditional residual terms as Re s is taken further negative. These residualterms contribute the two dominant bounds in our short-interval estimate.The first residual term ρ ( s ) has a pole which contributes O δ ( X /y ), whilethe rest of the residual terms contribute O δ,ǫ ( X y + ǫ (log y ) ) in aggregate.In Section 6.2 we then consider the contribution from the discrete spectraof Z ( s, , f × f ). Under the assumption y ≪ X − δ with δ > , the workof [HHR16] allows us to bound the discrete spectrum by O δ ( X y (log y ) ).Note that this estimate is slightly larger than the bound within the contin-uous spectrum. Lemma 2.5. πi Z ( σ s ) W ( s − f, f ) s + k − (cid:18) πs y (cid:19) X s y ds ≪ δ X y + X y (log y ) . Proof. See Section 6. (cid:3) Combining (2.9) with the three bounds from Lemmas 2.3, 2.4, and 2.5,we get the following theorem. HULSE, KUAN, LOWRY-DUDA, AND WALKER Theorem 2.6. For y ≪ X − δ with δ > , X | n − X | The Classical Conjecture for intervals of length X/y holdsfor any y such that the first error summand in (2.13) dominates. Ourerror bounds are thus optimized when y (log y ) = X . Unfortunately,the function y y (log y ) does not have an inverse expressible in com-mon functions; nevertheless, the minimal interval is well-approximated by | n − X | < X (log X ) .An application of summation by parts yields Theorem 1.1 as stated in theintroduction, which shows that the Classical Conjecture is true on averageover short intervals of length X (log X ) . In addition, we see that theHafner-Ivi´c bound, S f ( n ) ≪ n k − + , can be improved in the mean squareaspect over intervals of length slightly less than X . Corollary 2.8. For δ ∈ ( , ] , X δ X | n − X | In this paper, the assumption y ≪ X is essential to much of the analysis.There are two bounds in the paper that would need to be strengthened forTheorem 1.1 to hold in intervals of length X + ǫ .First, better subconvexity results for L ( + it, f × f ) would lead to bet-ter bounds for (6.11) in the continuous spectrum. Assuming the Lindel¨ofhypothesis for L ( s, f × f ), we get a bound for (6.11) consistent with theClassical Conjecture in intervals of length X + ǫ .Secondly, we bound (6.17) by absolute values and use X | t j |∼ T (cid:12)(cid:12)(cid:12) ρ j (1) h| f | Im( · ) k , µ j i (cid:12)(cid:12)(cid:12) ≪ T k +1 (log T ) , (3.1)from Proposition 4.3 of [HHR16], to estimate the contribution from the dis-crete spectrum in Section 6.2. There are good reasons to suppose that (3.1)is of the correct size, but in the process of taking absolute values we ignore HORT-INTERVAL AVERAGES OF SUMS OF FOURIER COEFFICIENTS OF CUSP FORMS9 oscillatory behaviour. We suspect it is possible to better take advantage ofthe oscillatory behaviour of the summands in (6.17) to lower the resultingbound.Progress towards the Classical Conjecture, such as in [Del74], [HI89],and [Ran90], has historically focused on full-integral weight cusp forms. It islikely that this restriction is unnecessary and that the analogue of the Clas-sical Conjecture will hold for half-integral weight cusp forms as well. Themethods in [HKLDW15] and this paper, with some small modifications,apply to the case of half-integral weight cusp forms.However, there is one important difference. The proof of (3.1) in [HHR16]relies heavily on f being of full-integral weight. A version of (3.1) has beenproven in the half-integral case in [Kir15], which takes the form X | t j |∼ T (cid:12)(cid:12)(cid:12) ρ j (1) h| f | Im( · ) k , µ j i (cid:12)(cid:12)(cid:12) ≪ T k + + ǫ . (3.2)Direct application of (3.2) along the lines of this paper leads to estimatesfor the discrete spectrum that deteriorate as the weight k increases. Witha bound more similar to (3.1), one could produce analogous results for half-integral weight forms in short intervals of length uniform in k .4. Properties of the Integral Transforms In this section, we prove the Mellin integral transform (2.6) and a gener-alization of that identity for later use in Section 7. Lemma 4.1. πi Z (2) exp (cid:18) πs y (cid:19) X s y ds = 12 π exp (cid:18) − y log X π (cid:19) . (4.1) Proof. Write X s = e s log X and complete the square in the exponents. Sincethe integrand is entire and the integral is absolutely convergent, we mayperform a change of variables s s − y log X/ π and shift the line ofintegration back to the imaginary axis. This yields12 πi exp (cid:18) − y log X π (cid:19) Z (0) e πs /y dsy . The change of variables s isy transforms the integral into the standardGaussian, completing the proof. (cid:3) Since the integral is absolutely convergent, we can produce additionalintegral identities by differentiating with respect to X under the integralsign. In particular, for non-negative integers m and ℓ , we multiply bothsides of (4.1) by X m + ℓ − , differentiate with respect to X a total of m times,and multiply by X − ℓ to get the following. Lemma 4.2. πi Z (2) Γ( s + m + ℓ )Γ( s + ℓ ) exp (cid:18) πs y (cid:19) X s y ds = X − ℓ d m dX m (cid:18) X m + ℓ − π exp (cid:18) − y log X π (cid:19)(cid:19) . By induction, it follows that the right-hand side above is O m,ℓ (cid:18)(cid:18) y + y | log X | (cid:19) m exp (cid:18) − y log X π (cid:19)(cid:19) , (4.2) where O m,ℓ indicates that the implicit constant depends on m and ℓ . Bounding the Contribution from W ( s ; f, f )In this section we prove Lemma 2.3. By separating the cases h = 0 and m = 0 and performing a small inclusion-exclusion correction, we collect W ( s ; f, f ) into a more standard Dirichlet series: W ( s ; f, f ) = X n ≥ a ( n ) S f ( n ) + a ( n ) S f ( n ) − | a ( n ) | n s + k − . (5.1)Denote the n th coefficient of this Dirichlet series by w ( n ). Since a ( n ) ≪ n k − + ǫ and S f ( n ) ≪ n k − + , by [HI89], we see that w ( n ) ≪ n k − + ǫ .Thus the Dirichlet series for W ( s ; f, f ) converges absolutely for Re s > + ǫ .By applying the integral transform (4.1) directly to W ( s ; f, f ), we get12 πi Z ( σ s ) W ( s ; f, f ) exp (cid:18) πs y (cid:19) X s y ds ≪ X n ≥ n + ǫ exp − y log ( Xn )4 π ! ≪ ǫ X + ǫ y . This completes the proof. Remark 5.1. This bound uses the best-known polynomial bound on S f ( n ),and so is probably weaker than reality; under the Classical Conjecture, thisterm would contribute no more than X + ǫ /y . Nevertheless, this term is somuch smaller than the others that improving the bound is not necessary forthis application.6. Bounding the Contribution from W ( s − f, f ) / ( s + k − πi Z ( σ s ) W ( s − f, f ) s + k − (cid:18) πs y (cid:19) X s y ds. (6.1)Recall that W ( s ; f, f ) = L ( s, f × f ) ζ (2 s ) − + Z ( s, , f × f ) and that Z ( s, , f × f ) splits into discrete and continuous components in its meromorphic con-tinuation as given in Proposition 2.2. HORT-INTERVAL AVERAGES OF SUMS OF FOURIER COEFFICIENTS OF CUSP FORMS11 In Section 6.1 we consider the part of (6.1) corresponding to L ( s, f × f ) ζ (2 s ) − and the continuous spectrum of Z ( s, , f × f ). In Section 6.2we consider the part of (6.1) corresponding to the discrete spectrum of Z ( s, , f × f ).6.1. Rankin-Selberg Convolution and Continuous Spectrum. Ac-counting for the residual terms that appear through meromorphic contin-uation, the contribution to (6.1) from the continuous spectrum of Z ( s − , , f × f ) / ( s + k − 2) is given by(4 π ) k πi ) Z ( σ s ) Z (0) (cid:20) Γ( s − + z )Γ( s − − z )Γ( s − s + k − ζ ( s − + z ) ζ ( s − − z ) ζ ∗ (1 + 2 z ) × hV f,f , E ( · , − z ) i + ⌊ − Re s ⌋ X m =0 ρ − m ( s − s + k − (cid:21) exp (cid:18) πs y (cid:19) X s y dzds. (6.2)Above, we are using the meromorphic continuation in (2.3). We refer to thefirst term in the brackets as the main integrand ; the remaining terms arethe residual terms . For m ≥ 0, the m th residual term ρ − m ( s − 1) appearsonly when Re( s − < − m . So the first residual term appears in themeromorphic continuation when Re s < .Now we move the line of s -integration in the negative direction. Fix δ > and suppose y ≪ X − δ . We shift σ to − M δ , where M δ > δ , which we specify later. During this shift, approximately M δ / ρ − m ( s − / ( s + k − 2) appear in the meromorphiccontinuation as indicated in (6.2) above. We separate each residual termand bound them separately from the shifted main integral.6.1.1. Shifted Main Integral. The identity for the Rankin-Selberg convolu-tion, as in Section 1.6 of [Bum98], implies hV f,f , E ( · , s ) i = 2 · Γ( s + k − π ) s + k − L ( s, f × f ) ζ (2 s ) . (6.3)We substitute (6.3) into the shifted main integrand of (6.2) and apply theasymmetric functional equation of the Riemann zeta function,2 ζ ( z )Γ( z ) = (2 π ) z sec( πz ) ζ (1 − z ) , (6.4)to the zeta functions in the numerator. Since Re s < 0, the reflected zetafunctions are in the domain of convergence and are uniformly bounded.Bounding the integrand by absolute values, we recall that 1 /ζ (1 ± z ) ≪| log z | for Re z = 0 (see [THB86, 3.11.10]), and apply Stirling’s approxima-tion to see that the integrand in (6.2) is O δ (cid:18) X − M δ | z | k − | log z || s | k − − M δ E ( s, z ) | L ( − z, f × f ) | (cid:19) , where E ( s, z ) are the collected exponential contributions E ( s, z ) = exp( π | Im s | )exp( π max {| Im s | , | Im z |} ) exp (cid:18) − π Im sy (cid:19) . To bound the integral, we split the line of z -integration into two regions.When | Im z | < | Im s | , we bound (6.2) by X − M δ Z ( − M δ ) | s | M δ log | s | exp (cid:18) − π Im sy (cid:19) Z | Im s |−| Im s | | L ( − it, f × f ) | dtdsy . The integral in the t -variable is quickly seen to be O ( | Im s | ) using Soundara-jan’s general weak subconvexity result [Sou10]. Soundarajan’s result indi-cates a further small fractional power of log | Im s | savings, which we neglectfor simplicity. The remaining integral is O δ (cid:0) X − M δ y M δ log y (cid:1) . (6.5)The contribution from the remaining region, | Im z | > | Im s | , is easierto bound as E ( s, z ) has exponential decay in both variables. In this case,Soundarajan’s subconvexity bound and a similar argument gives that thecontribution from this region is O δ (cid:0) X − M δ y M δ log y (cid:1) . (6.6)By choosing M δ large, the two bounds (6.5) and (6.6) can be made arbi-trarily small for X large enough, under the assumption y ≪ X − δ and δ > .While the implicit constant depending on M δ might be very large, it is in-dependent of X and y . In particular, if we choose M δ > δ − − (1 − δ ),the two bounds (6.5) and (6.6) are O δ (log y ) and non-dominant.6.1.2. The First Residual Term. The first residual term, ρ ( s − / ( s + k − Z ( s − , , f × f ) / ( s + k − 2) appears for Re s < . So beginning with σ s ∈ (2 , ) and substituting the definition (2.4) of ρ in the first residualterm in (6.2), we are led to consider the integral(4 π ) k πi Z ( σ s ) ζ (2 s − s − hV f,f , E ( · , − s ) i Γ( s − s + k − ζ ∗ (2 s − 4) exp (cid:18) πs y (cid:19) X s y ds. By expressing the inner product in terms of the Rankin-Selberg convolu-tion via (6.3), cancelling zeta functions, and applying the Gauss duplicationformula to the ratio Γ(2 s − / Γ( s − πi Z ( σ s ) Γ( s − )Γ( k + 2 − s ) L (3 − s, f × f )(4 π ) − s Γ( s − s + k − ζ (6 − s ) exp (cid:18) πs y (cid:19) X s y ds. (6.7)Shifting σ s into the interval ( , 2) passes over a pole at s = 2. It is shownin [HKLDW15] that the pole at s = 2 from L ( s − , f × f ) ζ (2 s − − iscancelled by the pole at s = 2 of Z ( s − , , f × f ). HORT-INTERVAL AVERAGES OF SUMS OF FOURIER COEFFICIENTS OF CUSP FORMS13 We then shift σ s to − M δ , passing approximately M δ poles at points ofthe form s = − m , with m ∈ Z ≥ . The residues at each pole are boundedin X and y by the pole with the largest residue, which occurs at s = . Thecontribution towards (6.7) from these polar terms is therefore O δ X y ! . (6.8) Remark 6.1. This is one of the two balancing terms that produces the finalbound in our main theorem.It remains to bound the shifted first-residual integral (6.7). As Re s < 2, we note that the L -function and ζ (6 − s ) are in their half-planes ofconvergence and are uniformly bounded. Writing s = − M δ + it , boundingthe integrand by absolute values, and applying Stirling’s approximation givesthat (6.2) is O δ (cid:18) X − M δ Z ∞−∞ | t | +2 M δ exp (cid:18) − πt y (cid:19) dty (cid:19) ≪ δ X − M δ y +2 M δ . (6.9)Just as in the analysis of the bounds (6.5) and (6.6), we can make (6.9)arbitrarily small as X tends to infinity by choosing M δ sufficiently large,provided that y ≪ X − δ with δ > . In particular, if we choose M δ > δ − − (1 − δ ), then (6.9) is non-dominant.6.1.3. Further Residual Terms. In [HKLDW15], it is shown that the secondresidual term of Z ( s − , , f × f ), which we denote as ρ ( s − 1) and whichappears for Re s < , identically cancels with L ( s − , f × f ) ζ (2 s − − . Inother words, the second residual term and the Rankin-Selberg convolution L -function completely cancel for Re s < .Further residual terms can all be handled systematically. For each integer m with 1 ≤ m ≤ M δ + , a further residual term ρ − m ( s ) / ( s + k − Z ( s − , , f × f ) / ( s + k − ρ − m ( s ) given in (2.5) intothe integral (6.2), we see that the integrals corresponding to these residualterms take the form2( − m (4 π ) k ζ ( − m ) m ! 2 πi Z ( σ m ) ζ (2 s + m − s + m − s − s + k − ζ ∗ (4 − m − s ) × hV f,f , E ( · , s + m − i exp (cid:18) πs y (cid:19) X s y ds, (6.10)where σ m = − m for each m . Notice that the apparent pole from Γ(2 s + m − 3) cancels with the pole of ζ ∗ (4 − m − s ) in the denominator, so thateach integrand is actually holomorphic along the lines of integration. Without assuming progress towards the Riemann Hypothesis, it is notpossible to shift the lines of integration further in the negative direction, asthe uncompleted Eisenstein series E ( · , s + m − 1) has poles at s = 1 + ρ − m for each nontrivial zero ρ of the zeta function.We therefore estimate each integral directly, writing the inner product interms of Rankin-Selberg convolutions via (6.3), reflecting the zeta functionin the numerator with its functional equation, bounding the integrand byabsolute values, and applying Stirling’s formula to bound each integral (6.10)by X − m Z ∞−∞ (1 + | t | ) m − | L ( + it, f × f ) || ζ (1 + 2 it ) | exp (cid:18) − πt y (cid:19) dty . (6.11)Recall that L ( s, f × f ) factors as the product L (Sym f, s ) ζ ( s ), where L (Sym f, s ) denotes the GL(3) symmetric square L -function. In [Li11], Lishows that | L (Sym f, + it ) | ≪ ǫ t + ǫ . Combined with the convexity bound | ζ ( + it ) | ≪ ǫ t + ǫ and de la Vall´ee-Poussin’s estimate, 1 /ζ (1 + it ) ≪ | log t | ,to show that the m th integral (6.11) is O ǫ,m ( X − m y m − + ǫ (log y ) ) . (6.12)Using once again the assumptions that y ≪ X − δ and δ > , we see thateach term of the form (6.12) is bounded above by the instance of (6.12) with m = 1. We have approximately M δ integrals, which jointly contribute O δ (cid:16) X y (log y ) (cid:17) . (6.13) Remark 6.2. This nearly matches the bound from 6.21, the second of thetwo balancing terms that produce the upper bound in Theorem 2.6. Notethat we have used the partial subconvexity bounds from [Li11], though bythe Lindel¨of hypothesis we should expect the vastly better bound L ( + it, f × f ) ≪ t ǫ .6.2. Discrete Spectrum. Beginning with σ s > , the contribution to-wards (6.1) from the discrete spectum of Z ( s − , , f × f ) / ( s + k − 2) isgiven by(4 π ) k X j ρ j (1) hV f,f , µ j i Z ( σ s ) Γ( s − + it j )Γ( s − + it j )Γ( s − s + k − × L ( s − , µ j ) exp (cid:18) πs y (cid:19) X s y ds. (6.14)Above, we are using the meromorphic continuation from Proposition 2.2. Apriori, the sum is over the complete orthonormal basis of Maass eigenforms,but we note that hV f,f , µ j i = 0 whenever µ j is odd, as was demonstrated in[HKLDW15] as a consequence of Watson’s triple product formula [Wat08].So we may assume that each µ j is even in the spectral sum above. HORT-INTERVAL AVERAGES OF SUMS OF FOURIER COEFFICIENTS OF CUSP FORMS15 For an even Maass form µ j , its corresponding L -function satisfies thefunctional equation [Gol06, Chap 3]Λ( s ) := π − s Γ (cid:18) s + it j (cid:19) Γ (cid:18) s − it j (cid:19) L ( s, µ j ) = Λ(1 − s ) , (6.15)indicating that L ( s, µ j ) has trivial zeroes at s = ± it j . So we shift the lineof integration in (6.14) to σ s = − ǫ without encountering any poles.Using the Gauss duplication formula, one can split the product of twogamma factors in the numerator of (6.14) into a product of four gammafactors, two of which are the necessary gamma functions to complete L ( s − , µ j ). Applying the functional equation (6.15) and the gamma reflectionproperty, Γ( z )Γ(1 − z ) = π csc( πz ), we find that all of these gamma factorscancel out.Up to constants and factors of 2 and π , which do not contribute to theupper bound, we are able to rewrite (6.14) as X j ρ j (1) hV f,f , µ j i Z ( σ s ) csc( π ( s − + it j )) csc( π ( s − − it j ))Γ( s − s + k − × L ( − s, µ j )(2 π ) s exp (cid:18) πs y (cid:19) X s y ds. (6.16)We then shift the line of integration from σ s = − ǫ to σ s = − M δ , where M δ is a constant depending on δ which we’ll still specify later. The integrandof (6.16) has poles at s = − m ± it j for m ∈ Z ≥ . The residues at thesepoles take the form1 y X j ρ j (1) hV f,f , µ j i csc( π ( − m ± it j )) L (2 + 2 m ∓ it j , µ j )Γ( − − m ± it j )Γ( k − m − ± it j ) × X − m ± it j (2 π ) ± it j exp π ( − m ± it j ) y ! . (6.17)Bounding the summands by absolute values and applying Stirling’s approx-imation, we bound (6.17) above by X − m y X j | ρ j (1) hV f,f , µ j i| | t j | m − k exp − πt j y ! . (6.18)In Proposition 4.3 of [HHR16], it is proved that X T ≤ t j ≤ T | ρ j (1) h| f | Im( · ) k , µ j i| ≪ T k +1 (log T ) / . (6.19) Weighting individual terms and noting that h V f,f , µ j i = 2 h| f | Im( · ) k , µ j i when µ j is even, it follows that X T ≤ t j ≤ T | ρ j (1) hV f,f , µ j i|| t j | m − k exp (cid:18) − πtj y (cid:19) ≪ T m (log T ) / exp (cid:18) − π ( T / y (cid:19) . Summing dyadically, we extend this bound to all t j to show that (6.18) is O m (cid:16) X − m y m (log y ) (cid:17) . (6.20)There are about M δ / m = 0 term in (6.20). It follows that the totalcontribution of these poles is O δ ( X y (log y ) ) . (6.21) Remark 6.3. This estimate is the second of the two balancing terms in ourfinal bound. We used absolute values to go from (6.17) to (6.18) in orderto directly apply (6.19), but this ignores the oscillatory behaviour of theoriginal summands.It remains to bound the shifted discrete component integral (6.16) on theline σ s = − M δ . Write s = − M δ + it , and split the integral and sum intotwo regions depending on the relative size of | t j | and | Im s | .When | t j | < | t | , bounding the integrand by absolute values and applyingStirling’s approximation leads to the upper bound ≪ δ X − M δ Z ∞ X | t j | < | t | (cid:12)(cid:12)(cid:12) ρ j (1) h| f | Im( · ) k , µ j i (cid:12)(cid:12)(cid:12) t M δ − k +3 exp (cid:18) − πt y (cid:19) dty . (6.22)Applying (6.19) shows that the inner spectral sum is O ǫ (cid:0) t k +1+ ǫ (cid:1) . Insertingthis bound gives the integral X − M δ Z ∞ t M δ +4+ ǫ exp (cid:18) − πt y (cid:19) dty . Performing the change of variables t ty removes y -dependence from theintegral, and we see that (6.22) is O ǫ,δ ( X − M δ y M δ + ǫ ) . (6.23)Similarly, when | t | < | t j | , we have the upper bound ≪ δ X − M δ Z ∞ X | t j | >t (cid:12)(cid:12)(cid:12) ρ j (1) h| f | Im( · ) k , µ j i (cid:12)(cid:12)(cid:12) exp (cid:18) π ( | t | − | t j | )2 (cid:19) × t M δ − k +3 exp (cid:18) − πt y (cid:19) dty . (6.24) HORT-INTERVAL AVERAGES OF SUMS OF FOURIER COEFFICIENTS OF CUSP FORMS17 Using the exponential decay and dyadic summation with (6.19), one canshow that the spectral parenthetical above is O ǫ ( t k +1+ ǫ ). Inserting thisbound into (6.24) and performing the change of variables t ty decouples X and y from the integral, and we have that (6.24) is also O ǫ,δ ( X − M δ y M δ + ǫ ) . (6.25)In both cases, the assumptions y ≪ X − δ and δ > indicate that (6.23)and (6.25) can be made arbitrarily small as X tends to infinity. In particular,if M δ > δ − − (1 − δ ), then these terms are non-dominant.Thus the total contribution from the discrete components of Z ( s, , f × f )is O δ ( X y (log y ) / ) . (6.26)7. Bounding the Contribution from the Mellin-BarnesIntegral (2.12)In this section, we prove Lemma 2.4 by bounding the double integral1(2 πi ) Z ( σ s ) Z ( σ z ) W ( s − z ; f, f ) ζ ( z ) B ( s + k − − z, z ) exp (cid:18) πs y (cid:19) X s y dzds, (7.1)in which B ( s, z ) = Γ( z )Γ( s )Γ( s + z )is the Beta function, σ z = ǫ ∈ (0 , σ s > + ǫ .Fix δ > y ≪ X − δ . Applying the asymmetric func-tional equation of the Riemann zeta function (6.4) allows us to rewrite (7.1)as 12(2 πi ) Z ( σ s ) Z ( − ǫ ) W ( s − z ; f, f ) ζ (1 − z ) × (2 π ) z Γ( s + k − − z )cos( πz )Γ( s + k − 1) exp (cid:18) πs y (cid:19) X s y dzds. (7.2)Note that the integrand in (7.2) experiences significant exponential decayin vertical strips, so we have absolute convergence away from poles. Wemay therefore simultaneously shift the two lines of integration, keeping σ s = + σ z + ǫ while shifting σ z to − ǫ − M δ , where M δ ≥ δ to be specified later. This introduces several poles, the first of whichcomes from ζ (1 − z ) as σ z passes by z = 0, and further poles at odd z = − m for 1 ≤ m ≤ M δ coming from the cosine term.The pole from ζ (1 − z ) at z = 0 has residue − πi Z ( + ǫ ) W ( s ; f, f )exp (cid:18) πs y (cid:19) X s y ds ≪ ǫ X + ǫ y . (7.3) The integral is a constant multiple of the term considered in Lemma 2.3,which gives the stated upper bound. The poles at negative odd integers z = − m for 1 ≤ m ≤ M δ each have residue r m πi Z ( − m + ǫ ) W ( s + m ; f, f ) Γ( s + k − m )Γ( s + k − 1) exp (cid:18) πs y (cid:19) X s y ds, (7.4)in which r m = ( − m ζ (1 + m )2(2 π ) m . Note that W ( · ; f, f ) lies within the half-plane of convergence of its Dirich-let series (5.1). Recall that w ( n ) denotes the n th coefficient of the Dirichletseries representation for W ( · ; f, f ). Expanding W ( · ; f, f ) in series and ap-plying the integral transform from Lemma 4.2 bounds (7.4) by O m (cid:18) r m X n ≥ w ( n ) n m + k − ( y + y | log( X/n ) | ) m exp (cid:18) − y log ( X/n )4 π (cid:19) (cid:19) . (7.5)Using the bound w ( n ) ≪ n k − + ǫ ′ as in Section 5, we bound (7.5) aboveby ≪ m,ǫ ′ Z ∞ t − m + ǫ ′ ( y + y | log( t/X ) | ) m exp (cid:18) − y log ( t/X )4 π (cid:19) dt. (7.6)The change of variables u = y log( t/X ) transforms (7.6) into X − m + ǫ ′ y m − Z ∞ e ( − m + ǫ ′ ) uy (1 + | u | ) m exp (cid:18) − u π (cid:19) du. When m = 1 and when y ≫ y = 1, Z ∞ e ( + ǫ ′ ) u (1 + | u | ) m exp (cid:18) − u π (cid:19) du, which converges independently of y . Similarly, for m > e ( − m + ǫ ′ ) uy , Z ∞ (1 + | u | ) m exp (cid:18) − u π (cid:19) du, which also converges independently of y . So we conclude that (7.5) is O m,ǫ ′ (cid:18) X + ǫ ′ (cid:16) yX (cid:17) m − (cid:19) for any ǫ ′ > y ≪ X − δ , the residue (7.4) associated to m = 1dominates the other residues for large X . There are at most M δ / O δ,ǫ ′ ( X + ǫ ′ ) . (7.7) HORT-INTERVAL AVERAGES OF SUMS OF FOURIER COEFFICIENTS OF CUSP FORMS19 Now, having shifted the lines of integration so that σ s = − M δ and σ z = − ǫ − M δ , we bound the shifted integral (7.2) in absolute value.The Dirichlet series W ( · ; f, f ) and ζ ( · ) are considered only within theirhalf-planes of absolute convergence. Using the Gamma reflection propertyΓ( z )Γ(1 − z ) = π csc( πz ) to raise the Gamma function in the denominatorto the numerator and applying Stirling’s approximation leads us to consider X − M δ Z ( σ s ) Z ( σ z ) | s − z | k + ǫ | s | M δ − k E ( s, z ) exp (cid:18) πs y (cid:19) dzdsy , (7.8)where E ( s, z ) are the collected exponential contributions, E ( s, z ) = exp (cid:0) − π | Im( s − z ) | − π | Im z | + π | Im s | (cid:1) . We bound (7.8) in cases, splitting the integral into intervals based on thesigns and relative sizes of Im s and Im z . The dominant contributions occurwhen E ( s, z ) provides no additional exponential decay, i.e. when Im s > Im z > s < Im z < 0. In this first case, let t = Im s and u = Im z and compute X − M δ Z ∞ Z t ( t − u ) k + ǫ t M δ − k exp (cid:18) − πt y (cid:19) dudty = O δ (cid:18) X y ǫ (cid:16) yX (cid:17) M δ (cid:19) . (7.9)The other cases are extremely similar and are also bounded by the errorterm in (7.9).Recalling that y ≪ X − δ , we see that (7.9) can be made arbitrarily smallfor large X , although perhaps with a very large implicit constant dependingon the size of M δ . In particular, if we choose M δ > δ − ( + ǫ ), then (7.9) isnon-dominant.In conclusion, by combining the upper bounds (7.3) and (7.7) and notingthat X + ǫ y − ≫ X + ǫ when y ≪ X − δ , we see that our initial integral (7.1)is indeed O δ,ǫ X + ǫ y ! , (7.10)as we set out to prove. ReferencesReferences [Bum98] D. Bump. Automorphic Forms and Representations , volume 55of Cambridge Studies in Advanced Mathematics . CambridgeUniversity Press, 1998.[CN62] K Chandrasekharan and Raghavan Narasimhan. Functionalequations with multiple gamma factors and the average orderof arithmetical functions. Annals of Mathematics , pages 93–136, 1962. [CN64] K Chandrasekharan and Raghavan Narasimhan. On the meanvalue of the error term for a class of arithmetical functions. Acta Mathematica , 112(1):41–67, 1964.[Del74] Pierre Deligne. La conjecture de Weil. I. Inst. Hautes ´EtudesSci. Publ. Math. , (43):273–307, 1974.[EH15] Anne-Maria Ernvall-Hyt¨onen. Mean square estimate for rel-atively short exponential sums involving Fourier coefficientsof cusp forms. In Annales Academiae Scientiarum Fennicae.Mathematica , volume 40, 2015.[Gol06] Dorian Goldfeld. Automorphic forms and L-functions for thegroup GL(n, R) , volume 13. Cambridge University Press, 2006.[HHR16] J. Hoffstein, T. Hulse, and A. Reznikov. Multiple Dirichletseries and shifted convolutions. Journal of Number Theory ,pages 457–533, 2016.[HI89] J. Hafner and A. Ivi´c. On sums of Fourier coefficients of cuspforms. L’Enseignement Math´ematique , :375–382, 1989.[HKLDW15] Thomas Hulse, Chan Ieong Kuan, David Lowry-Duda, andAlexander Walker. The second moment of sums of coefficientsof cusp forms. arXiv preprint arXiv:1512.01299v2 , 2015.[Jut87] M. Jutila. Lectures on a Method in the Theory of Exponen-tial Sums . Tata institute of fundamental research: Lectureson mathematics and physics. Tata institute of fundamentalresearch, 1987.[Kir15] E. Mehmet Kiral. Subconvexity for half integral weight L-functions. Mathematische Zeitschrift , 281(3-4):689–722, 2015.[Li11] Xiaoqing Li. Bounds for gl (3) × gl (2) l-functions and gl (3)l-functions. Annals of Mathematics , 173:301–336, 2011.[Ran90] RA Rankin. Sums of cusp form coefficients, automorphic formsand analytic number theory (Montreal, PQ, 1989), 115–121,univ. Montr´eal, Montreal, QC , 1990.[Sou10] Kannan Soundararajan. Weak subconvexity for central val-ues of L-functions. Annals of mathematics , 172(2):1469–1498,2010.[THB86] Edward Charles Titchmarsh and David Rodney Heath-Brown. The theory of the Riemann zeta-function . Oxford UniversityPress, 1986.[Wat08] Thomas C Watson. Rankin triple products and quantumchaos. arXiv preprint arXiv:0810.0425 , 2008.[Wu09] J Wu. Power sums of Hecke eigenvalues and application.