ON AN APPROXIMATE FUNCTIONAL EQUATION INVOLVING THE DIVISOR FUNCTION. 1. Introduction

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1 ON AN APPROXIATE FUNCTIONAL EQUATION INVOLVING THE DIVISOR FUNCTION ANNE-ARIA ERNVALL-HYTÖNEN Abstract. The error term in the approximate functional equation for exponential sums involving the divisor function will be improved under certain conditions for the parameters of the approximate functional equation. arxiv: v1 [math.nt] 22 Sep 2014 Exponential sums D 1, 2 ;α = 1. Introduction 1 n 2 dneαn with α R, ex = e 2πix, involving the divisor function dn = d n,d>01 are related to the exponential sums of Fourier coefficients of cusp forms. The main difference between the situations is that while the Fourier coefficients change signs, the divisor function is always positive. Therefore, one cannot obtain good general upper bounds. For instance D1,;0 = dn log, 1 n while the average estimate is much smaller: 1 2 dnenα dα = 0 1 n 1 n dn 2 log 3, and therefore, in average the estimate is log 3/2. The situation is different for the Fourier coefficients of holomorphic cusp forms: the upper bound n anenα 1/2 [5], is the same as the average estimate. The approximate functional equation involving the divisor function has been studied by Jutila [3], and even earlier by Wilton [10]. Jutila s result reads as follows: Theorem 1. Let 2, 1 1/2 be an integer, and 0 < η 2. Further assume h is a positive integer with gcdh, = 1 and with the property hh 1 mod. If 2 η 2 1, then for 1 < 2 2 we have D 1, 2 ; h +η = η 1 D 2 η 2 1, 2 η 2 2 ; h 2 η 1 +O 1/2 log. Currently the author is woring also for the University of Oulu. Approximately half of the wor was done while woring at the KTH in Stocholm on a grant no from Vetensapsrådet, and the rest of the wor supported by an Academy of Finland grants no and Finally, the author would lie to than the anonymous referee for thorough comments. 1

2 2 ANNE-ARIA ERNVALL-HYTÖNEN This is sharp because, for instance, when is a square, choosing =, h = 1 and η = 1, we have D, ; 2 +η log, and η 1 D η 2 2, 2 η 2 + 1, 1 1 = d1e However, under certain conditions, this result can be improved. The definition of a Farey sequence and a Farey approximation are needed to formulate the theorem. They can be found in Definition 4. Theorem 2. There exists an absolute const ant A 1 such that for any ε,ε > 0 there exists some a > 0 such that the following holds: Let c be an arbitrary positive real number; Let,,h be arbitrary integers satisfying 2, 1 1/4 and h, = 1, and let η be a real number satisfying η 1/4 1 and 2 η 2 clog A. For each j Z + set j = 2 η 3/2 1/2 2 η 2 ε 2 j, and let l be the smallest positive integer for which l c 1 2 η 3 2/5. For each j {1,2,...,l} we assume that the number β := h 2 η 1 has a Farey approximation h j j of order 1/2 ε j which satisfies β h j j c ε 1 j or j c 5/6 j 2 η 2 1/3. Then for any integers 1, 2 with 1 < 2 2 we have D 1, 2 ; h +η = η 1 D 2 η 2 1, 2 η 2 2 ; h 2 η 1 +O 1/2 2 η 2 a, for some positive constant a, where the implied constant depends only on ε,ε,c. Remar 3. The conditions for β seem technical and possibly fairly restricting. However, it will be noted that the set of β with 0 β 1 that do not satisfy either of these conditions is of Lebesgue measure at most O 1/4+ε. This article is strongly connected to [1]. These papers also have some overlapping statements. This was impossible to avoid without sacrificing readability. Generally, this article connects techniques from that paper, and from [5]. 2. Notation, preliminaries and nown lemmas From now on, let U = η 1/2 2 η 2 d with d a very small fixed positive number. Furthermore, we will assume that 1/4 and 2 η 2 1, since these are assumptions needed to prove the main result. To shorten the notation, write 1 = JU and 2 = + +JU with J a constant. Since 2 η 2 1, we have η 1/2 1/4 1/2, and hence U = 1/2 η 1/2 2 η 2 d 1/2+1/4 1/2 2 η 2 d 7/8 2 η 2 d = o, since d is very small. Throughout the paper, we assume the assumptions of Theorem 2. Furthermore, a smooth weight function w defined on some interval, say, on interval [,+ ] is a function satisfying the following conditions: 1 The support of w lies on the interval [, + ].

3 ON AN APPROXIATE FUNCTIONAL EQUATION INVOLVING THE DIVISOR FUNCTION 3 2 The function w satisfies the condition w j j for 0 j P for some P which typically depends on the ε needed. In practice, we will also require wx = 1 for all x on some suitable subinterval of [, + ]. Farey sequences and Farey approximations are needed throughout the article. They are defined in the following way: Definition 4. A Farey sequence of order Q consists of all rational numbers a b on some interval I in this paper I = [0,1] with the property gcda,b = 1 and 1 b Q. The Farey approximation of order Q is the approximation of a number as a sum a b +η, where a b belongs to a Farey sequence of order Q and η 1 bq. Write D 1, 2,α for the smoothed sum: D 1, 2,α = dneαnwn, 1 n 2 where w is a suitable smooth weight function. Sometimes it is easier to tae a main term out of the sum, and consider the rest: 1, 2,α = D 2 1, 2,α 1 logx+2γ 2logeηxwxdx. 1 This remaining part behaves a lot lie exponential sums of Fourier coefficients of holomorphic cusp forms. Actually, using the word "main term" is a bit misleading here as it often happens that the remaining term is actually larger than the main term. For instance, let us consider the following expression, where w is compactly supported on the interval [, + ], an satisfies the condition w j for 0 j P, where P is a sufficiently large positive number, and attains the value 1 on some subinterval of length :, + 3/4 4, 1/2 D, + 3/4, 1/2 1/2 4 see [2] but However, for instance + 1 logx+2γ 2logeηxwxdx 1. dn log +2γ 1, n and hence, in some very important cases this main term is also the actual, not just a symbolic, main term. The Voronoi type transformation formula see e.g. Theorem 1.7 [4] gives a nice expression for the term, + ;α: 1 D, + ;α = dn + 2πe + n h logx+2γ 2logwxeηxdx Y 0 4π nx +4 n h K 0 4π nx wxeηxdx For treating oscillating integrals the following lemma by Jutila and otohashi [6], Lemma 6 is very helpful:

4 4 ANNE-ARIA ERNVALL-HYTÖNEN Lemma 5. Let A be a P 0 times differentiable function which is compactly supported in a finite interval [a,b]. Assume also that there exist two quantities A 0 and A 1 such that for any non-negative integer ν P and for any x [a,b], A ν x A 0 A ν 1. oreover, let B be a function which is real-valued on [a, b], and regular throughout the complex domain composed of all points within the distance from the interval; and assume that there exists a quantity B 1 such that 0 < B 1 B x for any point x in the domain. Then we have b AxeBxdx A 0 A 1 B 1 P 1+ A P 1 b a. a For treating integrals which do not oscillate sufficiently fast, we need the saddlepoint lemma Lemma 3 in [5]. Before that we need some definitions. Definition 6. Define η J x such that it satisfies the following equation for any integrable funtion h: V V 2 2 V J u 2 du 1 du J hxdx = η J xhxdx 0 1 +u 1 0 where u = u u J and V < 2 1 /2J, and define η 0 to be the characteristic function of the interval [ 1, 2 ]. One may easily compute the Fourier transform ˆη J λ = iλ J+1 J e iλu 1 e iλ 2 e iλ 1. This implies that η J x is J 1 times differentiable. Denote by D the complex domain consisting of points z satisfying the condition z x < µ for some x [ 1, 2 ], where µ 1. Lemma 7 Saddle-point lemma. Let F 1 ε, G and V be positive constants. Let f be a real function such that f x F 1, f x > 0, f x F 2 1 for x [ 1, 2 ]. Let g be a holomorphic function with gz G in the domain D. Denote the characteristic function of 3 1, 1 +JV 2 JV, 2 by δx. Let x 0 be the possibly existing zero of f x + ι in the interval 1, 2, and suppose that V δx 0 F 1/2 1. Write E J x = G f x+ι +F 1/2 / 1 J+1.

5 ON AN APPROXIATE FUNCTIONAL EQUATION INVOLVING THE DIVISOR FUNCTION 5 Then 2 η J gxefx+ιxdx = ξ J x 0 gx 0 f x 0 1/2 efx 0 +ιx 0 +1/8 1 +O 1+δx 0 F 1/2 G 1 F 3/2 J +O V J E J 1 +jv+e J 2 jv, where ξ J is a bounded function on 1, 2 with the following properties: ξ J x = 1 on 1 +JV, 2 JV ξ J x is continuous and ξ J x V J on the set 3, except possibly at the points 1 +jv, 2 jv, j = 1,...,J 1. If x 0 does not exist, then the terms and conditions involving x 0 are to be omitted. j=0 3. Technical lemmas Lemma 8. Let η 1 1 ε, and wx be a smooth weight function satisfying the condition w j j for j P for a sufficiently large P. Then + 1 logx+2γ 2logeηxwxdx 1 log. Proof. Let us use partial integration P times to obtain + logx+2γ 2logeηxwxdx log 1 P η P 1. The following lemma will be formulated in the general case, but in practice it will be used when η is too small for the use of the previous lemma. Lemma 9. Let 6/5 2/5. Then + 1 logx+2γ 2logeηx dx 1/6 1/3+ε Proof. The proof is very simple. Just estimate using absolute values, and use the assumption for. Lemma 10. We have n h + 1 dne K 0 4π nx eηxdx 3/8. Proof. Let us first use the asymptotic bound for the K-Bessel function see e.g. [8] : π 1/2e K 0 z z. 2z

6 6 ANNE-ARIA ERNVALL-HYTÖNEN We have n h 1 dne + 1 1/2 1/4 This proves the lemma nx K 0 4π eηx dx n h dne + n ε 1/4 e 4π n 1/2 8 e 4π nx eηx dx nx 4 1/2 1/4 n 1/4 n 7/2 1/4 2 1/ /8 3/8. 4. Estimates for short sums We are interested in getting good estimates for short linear sums, but in order to get those, we need estimates for short non-linear sums. We will start by quoting a result which is probably not very well nown. Theorem 11. Let η,b R, and denote F = B 1/2. We assume that 2 F. Let Let g C 1 [, + ] and Then n + fz = ηz +Bz 1/2. g G, g G. dmgmefm 5/6 G+ G 1/2 F 1/3+ε. Proof. The full proof can be found in Karppinen s licentiate thesis [7]. However, as that particular wor is fairly difficult to come by, and available only in Finnish, I briefly tell what modifications have to be done to the proof of Theorem 4.1 in [2]. For simplicity, assume that gx = 1. Use Theorem 3.3 of [4]. ost of the terms can be treated similarly. However, with divisor function, we have to deal with a main term: F 1/2 logf 1 2/3 1/6 F 1/6+ε, r I where I corresponds the same Farey partitioning as in the original proof. The second difference is the error term arising from a change of a condition in the original theorem cusp forms versus the divisor function. This gives an error of size O 2/3 1/2 F 5/6+ε Finally, we may formulate the estimate for short linear sums. This theorem is similar to Theorem 5.1 [2], and there are very few differences in proofs. Theorem 12. Let 1, let the Farey approximation of order 1/2 ε 2 of a parameter α be α = h 1 + η, where η or 6/5 2/5, and ε 1 ε 2 is a sufficiently small fixed positive real number depending only on ε. Let wx be s smooth weight function defined on the interval [, + ] and satisfying the condition w j x j for 0 j P, where P is a large positive integer depending on ε.

7 ON AN APPROXIATE FUNCTIONAL EQUATION INVOLVING THE DIVISOR FUNCTION 7 1 If η 1+ε 2, then n + 2 Otherwise, further assuming we have n + dnenαwn 1/3+ε 1/6 η, dnenαwn 1/3+ε 1/6 + 1 η 1/2 1/2 2 η 2 ε. Proof. Proof is similar to [2], but requires the use of Lemmas 8 and 9 for treating the main term. Furthermore, the current second term 1 η 1/2 1/2 2 η 2 ε is slightly better than the original second term 1 η 1/2 1/2 ε. This improvement is already present in the formulation of the theorem in [1], and this comes from the fact, that this term corresponds to the case when there is one term in the Voronoi type summation formula that does not oscillate much, and in this case this term is just estimated using absolute values. Now, the Fourier coefficient of this term, say an 0 corresponds to some value of n 0 with n 0 2 η 2, and therefore, we have an 0 2 η 2 ε instead of the originally used weaer boundan 0 ε. Corollary 13. With the assumptions of the previous theorem, and the additional assumption 5/8, we have dnenαwn 1/3+ε 1/6. n + Proof. If η ε 1, then this is the special case of the previous theorem. If η 1+ε, then also η 1, and the result follows by a simple manipulation: 1 η 1/2 1/2 2 η 2 ε η 1/2 1/2+ε 3/2 1/2+ε 1/6 1/3+ε. The assumptions of the theorem and the corollary may loo fairly restricting, but actually, a very large proportion of all α [0,1] satisfy the conditions when = o 1 δ 1 for any fixed δ 1 > 0: The measure of the set of α which do no satisfy the conditions can be easily estimated. If α = h +η does not satisfy the conditions, then η 1 1 ε and 5/6 1/3. Now the measure of this set is for any given at most 1 ε, and now summing over the values of : 1 ε 5/6 1/3 2 ε 1 2/3 ε+2/3, 1 5/6 1/3

8 8 ANNE-ARIA ERNVALL-HYTÖNEN and hence, the measure of the set of α [0,1] not satisfying the conditions is at most ε+2/3 2/3, which is small when = o 1 δ 1 for any fixed positive δ1, because the ε can be required to be smaller than this δ 1. This corollary and Lemmas 8 and 9 together give an estimate for short smoothed sums notice that even though Lemma 9 is formulated for non-smoothed sums, this does not affect anything, because in the proof of the lemma, we only used absolute values. However, under certain conditions the weight function can be removed: Theorem 14. Let 5/8 and let α have Farey approximations h j j +η j of orders 1/2 ε, /2 ε, 1/2 ε,..., 2/5 δ 2 with η j j 1 ε or j 6/5 j 2/5 where j stands for 1 j 2. Then h dne +η 1/6 1/3+ε. n + Conditions again loo fairly technical and restricting. However, we may show that the measure of set of α [0, 1] not satisfying these conditions is small: We proved earlier that for any, the measure of the set of α [0,1] which do not satisfy the conditions, is O 2/3+ε 2/3. Using this estimate and dyadic intervals, we obtain that also here the total measure of the set of α not satisfying the conditions is O 2/3+ε 2/3. Since 5/8, this yields O 1/4+ε to be the total measure of the set for which the corollary does not hold. Proof. Proof is the similar to the proof of 5.5 [2]. However, we setch for sae of completeness. Also, some typos are fixed here. Let l > 0. Write 0 = = l = l 1 2, when l 2 and { 0 = l = l 1 5 i=1 i Consider the set of weight functions {w ±l 0 l L}, where L is large enough such that L 2/5+ε satisfying the following conditions: The support of the function is bounded: 1, when x [ , ] 0 4 w 0 x = 0, when x 0 or x and when l > 0, 1, when x [ l 1 + l 1, l+1 ] w l x = 0, when x l or x l + l. The derivatives are assumed to satisfy the following condition w j l x l j for 0 j J for some suitable value of J.

9 ON AN APPROXIATE FUNCTIONAL EQUATION INVOLVING THE DIVISOR FUNCTION 9 On the interval [ l+1, l + l ] the functions w l x and w l+1 add to one: w l+1 x = 1 w l x. Define l, l and w l x symmetrically with respect to the line x = + 2. The situation will loo lie the following: Use Corollary 13 to obtain the bound 1/6 i 1/3+ε on the interval [ i, i + i ]. Summing these bounds together gives the total bound 1/6 1/3+ε, as desired. Finally, define now a new function W 0 = w l. l Notice that n dnenα W 0 ndnenα 2/5+ε, which completes the proof of the theorem. Lemma 15. When 1/4, we have,h/ x 3/8+ε 1/4 +minx ε 2,x 1/2 c 2 for some positive constant c 2. Proof. Proof is similar to the proof of [5] Lemma 2. Theorem 16. For α = h +η with 1/4, η 1 1/4 and 2 η 2 b log for some 0 < b < 1 2 we have positive constants a and b such that for x x 1 η 1/2 1/2+b and x,x 1 Dx 1,x;α 1/2 2 η 2 a Notice that the condition 2 η 2 b log excludes thoseαfor whichη log 1/2b 1 1/2, and the total size of this set, whenever 1/4 is log 1/2b 1 1/2 = log 1/2b 1/2 1/4+ε, 1 1/4 1 h 1 1/4 so the estimate holds for almost all α. We are now ready to move to the proof. Proof. For simplicity. write Dt,α = n tdnenα.

10 10 ANNE-ARIA ERNVALL-HYTÖNEN We have 4 Dx 1,x;α = eηxd Write now x; h eηx 1 D x 1 ; h x 2πiη D t; h eηtdt. x 1 D t; h = 1 logt+2γ 1 2logt+E 0, h + t, h and similarly for D x; h and D x1 ; h. Hence Dx 1,x;α = eηx 1 logx+2γ 1 2logx+E 0, h + x, h eηx 1 1 logx 1 +2γ 1 2logx 1 +E 0, h + x 1, h x 2πiη 1 logt+2γ 1 2logt+E 0, h + t, h eηtdt Notice that x 1 eηx x; h x eηx 1 x 1 ;α 2πiη t; h eηtdt x 1 can be treated just lie in the proof of Lemma 4 in [5]. The same technique yields eηx x; h x eηx 1 x 1 ;α 2πiη t; h eηtdt 1/2 2 η 2 a for some positive constant a. Therefore, it suffices to wor with the main term and the E 0, h term. First of all, x 2πiη 1 tlogt+2γ 1 2log+E 0, h etηdt x 1 [ 1 tlogt+2γ 1 2logetη+E x 1 0, h ] x x etη + 1 logt+1etηdt x 1 x 1 = 1 xlogx+2γ 1 2logexη+ 1 x 1 logx 1 +2γ 1 2logex 1 η E 0, h x exη ex 1 η+ 1 logt+2γ 2logetηdt x 1 The first terms cancel out the main terms and the Esthermann zeta function term in 4, and therefore, it only remains to consider the integral x x 1 1 logt+2γ 2logetηdt. Now use the first derivative test to obtain the estimate x 1 logt+2γ 2logetηdt log 1/2 log 2 η 2 1/2 x 1 η which proves the theorem.

11 ON AN APPROXIATE FUNCTIONAL EQUATION INVOLVING THE DIVISOR FUNCTION Proof of the approximate functional equation Let us start with considering smoothed sums. Let w be a smooth weight function which satisfies the condition + +JU u U U + wxhxdx = U J du 1 du J hxdx JU+u 0 for any test function hx with u = u 1 + u u J. Recall U = η 1/2 1/2 2 η 2 d with d a very small fixed positive number. Also, J is a suitable large integer which will be defined later. The value of J will only depend on ε, and therefore, while some constants in some estimates will depend on the weight function, those constant can be chosen in such a way that they only depend on ε. Notice that since 2 η 2 1, we have U = η 1/2 1/2 2 η 2 d = η 1/2 1/2 1/2 1/4 1/2 2 η 2 d Now JU n + +JU = dnenαwn JU n 0 = 1/2 3/4 2 η 2 d 1/4 1/2 3/4 7/8. n + dnenαwn + JU n dnenα + n + +JU dnenαwn + dnenαwn dnenαwn. + n + +JU Theorem 16 gives the estimate 1/2 2 η 2 a for these sums. Therefore, we can use the smoothed sum instead of the non-smoothed one. Let us now use a Voronoi type summation formula 1 n dn + dnenαwn = 1 wxeηxlogx+2γ 2log dx+ 2πe n h Y 0 4π nx n h +4e K 0 4π nx wxeηxdx The first term, and the term containing the K-Bessel function have already been treated in the Lemmas 8 and 10 in Lemma 10, there is no weight function, but that does not matter: we get rid of the weight function in this case just by taing absolute values. Therefore, it is sufficient to concentrate on the term containing the Y-Bessel function. Let us write the Y-Bessel function as a sum of exponential terms see e.g. [8] : i 5 Y 0 z = 2πz 1/2 e z 1 4 π e z+1π 1 3/2 4 + e z 1 4 π +e z+1π 4 +O z 5/2 2πz

12 12 ANNE-ARIA ERNVALL-HYTÖNEN Substitute now z = 4π nx. The sum containing the integration over the last term the error term is very easy to treat: + 5/4 nx 1 dn dx = 1/4 5/4 We will now state some lemmas which are proved using partial integration, and which will be used later to handle the terms arising from the Voronoi-type transformation and the use of the asymptotic expansion of a Bessel function. The first term will yield the main contribution, and it is the most demanding to treat. The rest of the terms can be treated similarly, and they are much more straightforward. Therefore, we only briefly collect here the methods and contributions. 3/2 e z 1 4 π +e z+1 4 π can be estimated using Lemma 5, and the The part containing 1 2πz second derivative test [9], Theorem 5.9, and the contribution will be 1+ 2 η 2 ε. The sum containing the integration over the term i e z 1 2πz 1/2 4 π can also be treated by Lemma 5, and it gives the contribution 1. We may thus turn to the term yielding the main term. Substituting z = 4π nx and plugging in the first term in the asymptotic expansion in the place of the Y-Bessel function yields the expression + n h 1 i dn 2πe 2π4π nx = i n h + +JU dne n 1/4 2 n c 3 2 η 2 JU JU nx 1/2 e wxeηxdx 8 x 1/4 e 2 nx +ηx+ 1 wxdx 8 We first want to show that if c 3 is a large constant, then + +JU 1/2 dne n h n 1/4 wxx 1/4 e ηx 2 nx dx 1. To prove this bound, we first use Lemma 5 with Ax = wxx 1/4, Bx = ηx 2 nx A 0 = 1/4 1, A 1 = U, b a = +2JU, B 1 = nx wxx 1/4 e ηx 2 + +JU JU n and = 1. This gives dx 1/4 U P P P/2 n P/2 U + for any P J. Substituting this estimate, the left-hand side is dominated by 1/2 n ε 1/4 P/2 1/4+P/2 U P P U + n c 2 η 2 U + 1/2 1/4 2 η 2 ε 1/4 P 2 η 2 η 1/2 1/2 2 η 2 d, when P is sufficiently large. = U + 1/2 1/4 2 η 2 ε 1/4 2 η 2 d η P 1,

13 ON AN APPROXIATE FUNCTIONAL EQUATION INVOLVING THE DIVISOR FUNCTION 13 i 2 n c 3 2 η 2 n h + +JU dne n 1/4 JU x 1/4 e 2 nx +ηx+ 1 wxdx 8 Let us use the saddle point lemma to estimate the integral. The main term in the saddle point lemma only occurs if there is a saddle point term on the interval of integration. However, even if there is no saddle point, one can use the lemma. Let us now start with the case when there is a saddle point on the interval. Write fx = 2 nx. The saddle point is the point where n f x+η = +η = 0, x that is, when x 0 = n. This point is on the interval when 2 η 2 JU n 2 η 2 + +JU 2 η 2 JU n 2 η 2 + +JU. We will first calculate the main term in the saddle point lemma. Notice that and nx0 fx 0 +ηx 0 = 2 f x 0 = The saddle point lemma yields + +JU x 1/4 e 2 nx +ηx+ 1 8 JU = wn 2e J +O 1/4 U J n 2x 3/ n n 1/4 η 2 j=1 +ηx 0 = n η 2 = η3 2 2n n wxdx = wn 2 η 2 η η +O j=1 1/4 η 3 2 2n +δn1/4 n 3/4 n 3/2 n ju + n1/4 3/4 J 1 1/2 1 e 8 n η 2 J J 1 n +O 1/4 U J η + n1/4, + +ju 3/4 where wn = 0 and δn = 0 if n 2 η 2 JU or n 2 η 2 + +JU wn = 1 and n = 1 if 2 η 2 n 2 η 2 + wn 1, w n 2 η 2 U 1 andδn otherwise We may now calculate the sum over the main terms: i n h wndne 2 2 η 2 JU n 2 η 2 + +JU = 1 η 2 η 2 JU n 2 η 2 + +JU 1 n 2e 1/4 8 n n 1/4 η 2 η n h wndne n η 2

14 14 ANNE-ARIA ERNVALL-HYTÖNEN When 2 η 2 n 2 η 2 +, ie. when wn = 1, this yields the main term in the approximate functional equation. When n is not on this interval, these sums contribute to error terms. Estimating them is rather simple. Since the length of the sum can be estimated to be 2 η 2 U = 2 η 2 2 η 2 d η 1/2 1/2 = 2 η 2 d+1 η 1/2 2 η 2 1/2+d 2 η 2 5/8, and therefore, we may use Theorem 14 and partial integration to estimate the sum: 1 n h wndne n η η η 2 1/3+ε 2 η 2 U 1/6 η 2 η 2 JU n 2 η 2 1/2 2 η 2 ε+d/6 1/12. The sum on the interval [ 2 η 2 +, 2 η 2 + +JU] can be estimated similarly. We still need to treat the error terms arising from the use of the saddle point lemma. The first error term is very easy to estimate: 1/2 n ε 1/4 3/2 +δn1/4 n 3/4 2 η 2 ε+d n 1 n 2 η 2 +δ+ju The sums containing the two other error terms are estimated in Lemma 2.8 in [1] by chopping the sums into three pieces depending on which part of the error term is the dominating one. This yields a total error of 2 η 2 ε Jd. Since we may choose J as large as we wish to ensure that ε Jd < a, this proves the theorem. References [1] A.-. Ernvall-Hytönen. On the error term in the approximate functional equation for exponential sums related to cusp forms. International Journal of Number Theory, 45, [2] A.-. Ernvall-Hytönen and K. Karppinen. On short exponential sums involving Fourier coefficients of holomorphic cusp forms. Int. ath. Res. Not. IRN, 10:Art. ID. rnn022, 44, [3]. Jutila. On exponential sums involving the divisor function. J. Reine Angew. ath., 355: , [4]. Jutila. Lectures on a ethod in the Theory of Exponential Sums, volume 80 of Tata Institute of Fundamental Research Lectures on athematics and Physics. Published for the Tata Institute of Fundamental Research, Bombay, [5]. Jutila. On exponential sums involving the Ramanujan function. Proc. Indian Acad. Sci. ath. Sci., 971-3: , [6]. Jutila and Y. otohashi. Uniform bound for Hece L-functions. Acta ath., 195:61 115, [7] K. Karppinen. Exponential Sums Relating to Cusp Forms Finnish. University of Turu, Licenciate thesis. [8] N. N. Lebedev. Special Functions and Their Applications. Dover Publications Inc., New Yor, Revised edition, translated from the Russian and edited by Richard A. Silverman. [9] E. C. Titchmarsh. The Theory of the Riemann Zeta-Function. Oxford, at the Clarendon Press, [10] J. R. Wilton. An approximate functional equation with applications to a problem of Diophantine approximation. J. Reine Angew. ath., 169: , Department of athematics and Statistics, University of Helsini, Finland