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The twelfth moment of central values of Hecke series
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Journal of Number Theory 108 (2004) 157–168
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The twelfth moment of central values of Hecke series Matti Jutila Department of Mathematics, University of Turku, FIN-20014 Turku, Finland Received 2 October 2003; revised 18 March 2004
Communicated by B. Conrey
Abstract Let Hj ðsÞ be the Hecke L-function attached to the Maass wave form for the jth eigenvalue 14 þ k2j of the hyperbolic Laplacian acting in the Hilbert space of automorphic functions for the full modular group. The following mean value estimate for the central values Hj ð12Þ is proved: X 1 5K 4þe : Hj12 2 kj pK r 2004 Elsevier Inc. All rights reserved. Keywords: Maass forms; Hecke series; Central values
1. Introduction This paper is related to [6], where the fourth moment of the central values Hj ð12Þ of the Hecke series Hj ðsÞ ¼
N X
tj ðnÞns
n¼1
Fax: +358-2-3336595. E-mail address: jutila@utu.fi.
0022-314X/$ - see front matter r 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2004.05.012
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over a short spectral interval was estimated. We adopt the notation of the monograph [11] of Motohashi; thus the coefficients tj ðnÞ are the Hecke eigenvalues of the Maass wave form attached to the jth eigenvalue lj ¼ 14 þ k2j of the hyperbolic Laplacian. The Fourier coefficients rj ðnÞ of the form in question are connected with the Hecke eigenvalues by the relation rj ðnÞ ¼ tj ðnÞrj ð1Þ; and we write aj ¼ jrj ð1Þj2 =coshðpkj Þ as usual. In this notation, the main result of [6] was X 1 aj Hj4 5K 4=3þe ; 2 1=3
ð1:1Þ
jkj KjpK
actually, as a slight improvement (see [4, Eq. (4.12)]), the factor K e here can be due to Iwaniec [5], this replaced by ðlog KÞ16 : Together with the estimate aj bke j implied the ‘‘subconvexity’’ bound 1 1=3þe Hj : ð1:2Þ 5kj 2 This had been proved previously by Ivic´ [3] as a corollary of his estimate X 3 1 aj H j 5K 1þe 2 jk Kjp1
ð1:3Þ
j
and the non-negativity of Hj ð12Þ: Now, in view of (1.1) and (1.3), sums of the mixed form X 4 1 aj H j F ðxÞ ¼ 2 xpk pxþ1 j
for x^K are of interest. An optimal estimate would be F ðxÞ5K 1þe in analogy with (1.3), whereas (1.1) implies the unconditional bound F ðxÞ5K 4=3þe : The statistics of the values of F ðxÞ will be studied in terms of the mean square Z 2K SðKÞ ¼ F 2 ðxÞ dx: ð1:4Þ K
Also, let SðK; UÞ denote the same integral restricted to values of x satisfying F ðxÞXU: Theorem 1. We have SðKÞ5K 3þe
ð1:5Þ
SðK; UÞ5K 4þe U 1 :
ð1:6Þ
and
As a corollary of (1.6), we get (see Section 4) an estimate for the twelfth moment of Hj ð12Þ:
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Theorem 2. We have X kj pK
Hj12
1 5K 4þe : 2
ð1:7Þ
This can be viewed as an analogue of the well-known estimate Z
T
0
12 1 z þ it dt5T 2þe 2
ð1:8Þ
of Heath-Brown [2] which in fact will enter in the course of our argument. Also, note that (1.7) implies (1.2) as an immediate corollary and gives thus a third proof for the latter. As a novelty compared with (1.1) and (1.3), Theorem 2 shows that exceptionally large values of Hj ð12Þ can occur only very rarely. The proof of Theorem 1 will be based on a functional equation for weighted spectral sums of products Hj ðz1 ÞHj ðz2 ÞHj ðz3 ÞHj ðz4 Þ; where the z-points lie on or close to the critical line. On the other side of that functional equation, a similar sum appears with another quadruple of points and another spectral weight function. Such a curious relation, analogous to transformations of exponential sums by Poisson’s or Voronoi’s sum formula, was first observed by Kuznetsov [8] though essential corrections, due to Motohashi [12], were required in the formulation and proof of his claim. However, if the ‘‘parity sign’’ ej ¼ 71 of the jth Maass form is inserted as a factor to the product, then the problem becomes essentially easier. On the other hand, if at least one of the numbers zj equals 12; then this modification is harmless, for it follows from the functional equation of Hj ðsÞ that Hj ð12Þ ¼ 0 if ej ¼ 1: This observation was utilized by Motohashi in the context of his identity for sums of the type N X j¼1
aj tj ð f Þhðkj ÞHj2
1 2
(see [11, Lemma 3.8]), and this identity was the starting point in the proofs of (1.1) and (1.3). In the case of the spectral sum Hj4 ð12Þ; the quadruple ð12; 12; 12; 12Þ remains invariant in the transformation process and only the weight function is changed. We encountered this phenomenon in [6] by an ad hoc argument. Actually we did not get a functional equation strictly in the above sense, for the role of the power Hj4 ð12Þ was played by the product of Hj2 ð12Þ with a finite sum representing approximately another factor like this. An analogous phenomenon occurred in [7], where a spectral sum over jHj ð12 þ itÞj4 was transformed into another sum involving products Hj3 ð12ÞHj ð12 2itÞ: Turning to the method of the present paper, the proof of (1.5) will start (as in [6]) from a short spectral sum of aj Hj4 ð12Þ with a Gaussian weight. If the relevant values of kj in the original sum lie near x; then the weight function in the transformed sum is
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an oscillating function of x: When this relation is squared and integrated over xA½K; 2K ; a spectral double sum appears on the transformed side, and only those terms lying near the diagonal will be significant. Then, we get integrals of the type SðKÞ on both sides of the relation, but the new spectral size parameter—playing the role of K—will be smaller than K; the result is the inequality (3.13). Iterating this procedure, we end up eventually with a sufficiently short integral which can be estimated trivially without essential loss of accuracy, and the claim (1.5) follows. For a proof of (1.6), we combine the transformation device with the estimate (1.5) already established. Though the above argument could be carried out on the basis of the method of [6], it is more convenient to refer to an approximate variant of the fundamental functional equation given in [9] or [4]; the latter paper contains corrections and simplifications to the presentation in the former one. Also, an approach to this functional equation is outlined by Motohashi [12] as an easier variant of his sophisticated argument therein, and the details can be readily supplied following his guidelines.
2. The basic functional equation The following lemma of Kuznetsov ([9, Lemma 4.7]) is an approximate and specialized version of the above-mentioned functional equation. Lemma. Let K be a large parameter, 0oao1=2; and K a pQpK 1=2 : Then for x^K we have ! ! Z 8 kj x 2 1 2 N jzð12 þ irÞ rx 2 exp exp þ dr Q 2 p 0 jzð1 þ 2irÞj2 Q jX1 Z 8 X 1 1 N jzð12 þ irÞj aj Hj4 hðr; xÞ dr ¼ xQP6 ðlog xÞ þ hðkj ; xÞ þ 2 p 0 jzð1 þ 2irÞj2 jX1
X
aj Hj4
þ OðQ2 log6 KÞ:
ð2:1Þ
Here P6 is a polynomial of degree 6, and the function hðr; xÞ can be represented, for 15r5x; by an asymptotic expansion in 1=r with leading term of the form h0 ðr; xÞ ¼ bðr; xÞQr1=2 expðQ2 Z2 ðr=xÞ=4Þ sinðrZðx=rÞ þ xZðr=xÞ þ p=4Þ;
ð2:2Þ
where ZðyÞ ¼ 2 arsinh y and bðr; xÞ is a smooth stationary bounded function, i.e. @ i @ j bðr; xÞ 5i;j ri xj for i; jX0: @ri @x j
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Continuing the comments made in the introduction, let us further discuss the nature of the above functional equation and its proof. As we pointed out already, a somewhat less explicit transformation formula of the same flavor follows from the calculations in Section 3 of [6]. Indeed, in that paper, the weight function on the transformed side is essentially a factor of the order QK 1=2 times the real part of Z pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G2 ð12 þ irÞ22ir N bðyÞyir ð y1 þ 1 þ y1 Þ2ix Gð1 þ 2irÞ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 þ 1 þ y1 Þ2ir expðQ2 log2 ð y1 þ 1 þ y1 ÞÞ dy;
ð2:3Þ
where bðyÞ is stationary function of order y5=4 : The saddle point of the y-integral is ðx=rÞ2 ; and the saddle-point method yields a weight function of the form (2.2). Turning to a sketch of the proof of the lemma, one may start from the sum N X
ej aj Hj ðz1 ÞHj ðz2 ÞHj ðz3 ÞHj ðz4 Þ:
ð2:4Þ
j¼1
To begin with, letting the numbers zi lie in the domain of absolute convergence, this sum is expressed in terms of Kloosterman sums (up to some less significant terms) by use of the Spectral-Kloosterman sum formula of Bruggeman and Kuznetsov, next the resulting arithmetic sum is transformed by the functional equation of the Estermann zeta-function to another sum involving again Kloosterman sums, and finally the latter expression is translated back into the language of spectral theory by Kuznetsov’s Kloosterman-Spectral sum formula. The spectral expression for the sum (2.4) allows its analytic continuation to the central point with all zi ¼ 12; where the factors ej become irrelevant. This argument is indicated in [12], where the main focus, however, is the more difficult sum without the parity signs. The underlying reason for the difference between these two cases (i.e. with or without the parity signs) is that these lead, respectively, to the ‘‘opposite-sign’’ or ‘‘equal-sign’’ case of the Spectral-Kloosterman sum formula (see Theorems 2.2 and 2.4 in [11], and Eqs. (2.15) and (2.17) in [12]). In the former case, this formula involves the integral transform Z N Gðs þ irÞ rhðrÞ dr; ð2:5Þ N Gð1 s þ irÞ cos ps while in the latter case the corresponding transform is Z N Gðs þ irÞ rhðrÞ dr: N Gð1 s þ irÞ cosh pr
ð2:6Þ
An apparent advantage of the transform (2.5) over (2.6) is its exponential decay as a function of Im s:
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The above argument leads to a transformation formula with a certain weight function attached to Hj4 ð12Þ on the transformed side. The dominating part of the weight function is Z 1 4 1 s coshðprÞ sin2 ðpsÞjðsÞ ds; Gðs þ irÞGðs irÞG ð2:7Þ 2p5 i ðdÞ 2 where d is a small positive number and Z N Gðs iuÞGðs þ iuÞhðuÞu sinhðpuÞ du jðsÞ ¼ N
with ( ! !) ðu2 þ 1=4Þðu2 þ 9=4Þ ux 2 uþx 2 hðuÞ ¼ 2 exp þ exp : ðu þ 1=4Þðu2 þ 9=4Þ þ 626 Q Q To analyze the integral (2.7), one may use the formula (see [11, pp. 119–120]) Z 1 G2 ð12 sÞGðs þ irÞGðs irÞ sinðpsÞxs ds 2pi ðdÞ ( G2 ð12 þ irÞ 1 p F ð þ ir; 12 þ ir; 1 þ 2ir; 1=xÞxir ¼ 2 coshðprÞ Gð1 þ 2irÞ 2 ) G2 ð12 irÞ 1 ir F ð ir; 12 ir; 1 2ir; 1=xÞx : þ ð2:8Þ Gð1 2irÞ 2 We apply now the following property of Mellin transforms (which is a corollary of Theorem 73, that is Parseval’s formula, in [13]): if the Mellin transforms of the functions f ðxÞ and gðxÞ are F ðsÞ and GðsÞ; respectively, and the functions f ðxÞxc1=2 ; gðxÞxc1=2 lie in the Hilbert space L2 ð0; NÞ; then Z Z N 1 F ðsÞGðsÞ ds ¼ f ð1=xÞgðxÞx1 dx: 2pi ðcÞ 0 Writing the right-hand side of (2.8) as f ðx; rÞ for short, we see that the Mellin transform of f ð1=x; rÞ is 1 s Gðs þ irÞGðs irÞ sinðpsÞ; G2 2 and we may choose c ¼ d as in (2.7). We apply next the above formula to the pair f ðx; rÞ; f ðx; uÞ to write the integral (2.7) in terms of hypergeometric functions. Then, if the hypergeometric functions are transformed by the formula (see [10,
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Eq. (9.6.12)]) F ða; b; 2b; zÞ ¼
1þ
pffiffiffiffiffiffiffiffiffiffiffi!2a 0 pffiffiffiffiffiffiffiffiffiffiffi!2 1 1z 1 1 1 1z pffiffiffiffiffiffiffiffiffiffiffi A F @a; a b þ ; b þ ; 2 2 2 1þ 1z
and the u-integral is calculated, we end up with an integral of the type (2.3), and as we pointed out above, this yields the formula (2.2). In the course of these calculations, the new hypergeometric function can be expressed simply by the hypergeometric series if it converges rapidly, or else by the Gaussian integral. We remarked in the introduction that functional equations such as (2.1) or its generalizations can be viewed as analogues of the ‘‘Process B’’ in van der Corput’s method, that is transforming exponential sums by Poisson’s summation formula, and in view of the preceding discussion the analogy is indeed not only formal. The role of the Poisson summation is now played by the functional equation for Estermann’s functional equation, equivalent to the Voronoi summation. The original spectral sum is first transformed into an arithmetic form as a preparation for the ‘‘reflection’’ step—the core of the argument—after which the new arithmetic expression is again interpreted in terms of spectral theory. A methodical question naturally arising in the context of the proof of the functional equation is the curious role of the Kloosterman sums: why do these first appear and then disappear, without leaving any trace; does it mean that they are redundant, after all? Indeed, it was pointed out to the author by Prof. Motohashi that there ought to be an alternative and more functional approach to the functional equation avoiding Kloosterman sums, and it is our aim to return jointly to this topic elsewhere. To this end, one should interpret the functions Hj ðsÞ as L-functions attached to irreducible automorphic representations of the group G ¼ SLð2; RÞ rather than to particular automorphic forms, and similarly all functions of interest should ‘‘live’’ in the group G rather than in the upper half-plane. An excellent example of a methodical recasting like this is the recent paper [1] by Bruggeman and Motohashi, where a new proof for Motohashi’s spectral formula for a weighted fourth moment of Riemann’s zeta-function is given dispensing with the spectral theory of sums of Kloosterman sums.
3. Proof of Theorem 1 We prove first the estimate (1.5), which will then be needed in the proof of (1.6). We decompose the integral (1.4) defining SðKÞ according to the size of F ðxÞ: The contribution of the values x with F ðxÞpK 1þd to SðKÞ is 5K 3þ2d : Here d is a small positive number which will be fixed during the following discussion, and e is another small positive number which can be chosen sufficiently small in relation to d:
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Next we consider, for UXK 1þd ; the contribution, say S ðK; UÞ; of the values x such that UpF ðxÞp2U:
ð3:1Þ
Q ¼ UK 1e log K
ð3:2Þ
Put
with 0oeod; and cover the interval ½K 1; 2K by intervals of the form IðPÞ ¼ ½P; P þ Q : Let N be the set of the integers n such that (3.1) holds for some xA½n; n þ 1 : We classify the intervals IðPÞ according to the cardinality of the set N -IðPÞ: For a parameter JX1; let RðJÞ be the number of those intervals IðPÞ for which this cardinality lies in the interval ½J; 2JÞ: Then X S ðK; UÞ5 JRðJÞU 2 ; ð3:3Þ J
where J runs over the dyadic numbers 2n ; n ¼ 0; 1; 2; y : The number RðJÞ is now estimated by use of (2.1), which we write as AðxÞ ¼ xQP6 ðlog xÞ þ BðxÞ þ CðxÞ þ OðQ2 log6 KÞ
ð3:4Þ
for short. If IðPÞ is one of the RðJÞ intervals selected above, then AðxÞbJU for xAIðPÞ; note that the second term on the left-hand side of (2.1) is non-negative. By (3.2), the quantity AðxÞ ðbUÞ dominates over the first and last term in (3.4), so that AðxÞ5jBðxÞj þ jCðxÞj: Moreover, since hðr; xÞ5Qr1=2 ; we may omit the terms with kj pK e in BðxÞ: Let ˜ BðxÞ denote the remaining sum, where the leading term of the asymptotic expansion for hðkj ; xÞ is now given by the formula (2.2). We square the above-mentioned ˜ inequality, with BðxÞ replaced by BðxÞ; and integrate over the RðJÞ intervals IðPÞ: For an upper estimation of the integral arising on the right, we introduce a weight function wðxÞ; which is a smoothed version of the characteristic function of ½K; 2K : Then Z Z 2 ˜ RðJÞQðJUÞ2 5 jBðxÞj wðxÞ dx þ jCðxÞj2 wðxÞ dx: ð3:5Þ Consider next the integral Z X 2 ˜ wðxÞ dx ¼ jBðxÞj kj ;kk 4K e
aj ak Hj4
Z 1 1 Hk4 wðxÞhðkj ; xÞhðkk ; xÞ dx: 2 2
ð3:6Þ
Here, we replace h by the leading term h0 noting that the other terms give analogous but smaller contributions. The exponential factor in (2.2) truncates the spectral sums
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in (3.6) to the following finite double sum: K e okj ; kk pk ¼ KQ1 log K ¼ K 2þe U 1 pK 1dþe pK 1d=2
ð3:7Þ
if we suppose that eod=2: The integrals on the right-hand side of (3.6) depend on the oscillatory nature of the integrand, that is on the sine factor, say sinðjðr; xÞÞ; in (2.2). Then jðr; xÞx ¼ arsinhðr=xÞ; and hence jjðr; xÞxr j^K 1 for the relevant values of the variables. Thus, the integrand on the right-hand side of (3.6) is oscillatory if jkj kk jb1; and the double sum can be truncated to jkj kk j5K e ; for otherwise repeated integration by parts shows that the integral is negligibly small. On the other hand, if kj A½r; r þ 1 and kk A½s; s þ 1 with jr sj5K e ; then the integral in (3.6) can be estimated by 5KQ2 r1 ð1 þ jr sjÞA
ð3:8Þ
for any fixed large number A: Since this depends only on r and s but not on kj and kk ; the summations over these pairs ðkj ; kk Þ in (3.6) can be separated in the upper estimation. Next note that X X 1 1 1 2 4 4 aj H j ak Hk p ðF ðrÞ þ F 2 ðsÞÞ: 2 2 2 spkk osþ1 rpkj orþ1 Multiplying this by the expression (3.8), integrating over ðr; sÞA½K e ; k ½K e ; k with s ¼ r þ OðK e Þ; and recalling the choice (3.2) of Q; we obtain Z
2 ˜ jBðxÞj wðxÞ dx5K 12e U 2 log2 K
Z
k
r1 F 2 ðrÞ dr;
1
the reappearance of the function F is an important phenomenon here. Hence Z
2 ˜ jBðxÞj wðxÞ dx5K 12e U 2 log3 K max L1 SðLÞ:
ð3:9Þ
Lpk
Similarly, noting that jzð1 þ 2irÞj1 5log r for rX2; we have Z
2
jCðxÞj wðxÞ dx5K
12e
2
6
Z
k
U log K 1
16 1 r z þ ir dr: 2 1
By (1.8), the standard estimate for the zeta-function, and (3.7), this is further 5K 1ð5=3Þe U 2 k5=3 ¼ K 7=3 U 1=3 :
ð3:10Þ
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Combining now (3.3), (3.5), (3.9), and (3.10), we get X J 1 RðJÞQðJUÞ2 S ðK; UÞ5 Q1 J
5 K U log2 K max L1 SðLÞ þ K 10=3þe U 2=3 : e
Lpk
ð3:11Þ
Turning to the estimation of SðKÞ; let U run over the numbers 2n K 1þd for n ¼ 0; 1; 2; y and note that X SðKÞ5K 3þ2d þ S ðK; UÞ: U
Thus SðKÞ5K 3þ2d þ S ðK; UÞ log K
ð3:12Þ
for some UXK 1þd : Since the last term on the right-hand side of (3.11) is dominated by the first term on the right-hand side of (3.12), we obtain SðKÞ5K 3þ2d þ K e U log3 K max L1 SðLÞ: Lpk
This implies that SðKÞ SðLÞ 51 þ max 3þ2d ðL2þ2d K 32de U log3 KÞ: Lpk L K 3þ2d By (3.7) and the assumption UXK 1þd ; the last factor on the right-hand side is 2
5K ð2þ2dÞð2þeÞ32de U 12d log3 K5K dþeþ2ed2d log3 K51: Thus, since kpK 1d=2 ; we conclude that SðKÞ SðLÞ 51 þ max 3þ2d 1d=2 K L3þ2d 1pLpK
ð3:13Þ
and (1.5) follows if this relation is iterated sufficiently many times and the quotient occurring at the last stage is estimated trivially. For a proof (1.6), we proceed as above with slight modifications. We let now e stand generally for a small positive number, not necessarily the same at each occurrence. For U5K 1þe ; the estimate (1.6) follows from (1.5). Otherwise K 1þe 5U5K 2þe by the ‘‘trivial’’ estimate F ðxÞ5x2þe ; and by (3.11), (1.5), and (3.7) we have S ðK; UÞ5 ðk2 U þ K 10=3 U 2=3 ÞK e 5 K 4þe U 1 ð1 þ U 1=3 K 2=3 Þ5K 4þe U 1 :
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This completes the proof of (1.6), since the same estimate follows for SðK; UÞ as well by summation over dyadic values of U: &
4. Proof of Theorem 2 The contribution of the terms with Hj ð12ÞoK 1=4 to the sum (1.7) is 5K 4þe in view of the known estimate for the fourth moment of Hj ð12Þ: We classify the remaining terms according to the condition V pHj ð12Þo2V ;
ð4:1Þ
where V XK 1=4 runs over dyadic values. For nA½K; 2KÞ; either the condition (4.1) holds for no spectral parameter kj A½n; n þ 1 ; or else there is a non-negative integer n such that this condition holds for at least 2n and less than 2nþ1 parameters. We now estimate the contribution of those terms corresponding to a certain value J ¼ 2n ; let TðJÞ be the number of the relevant values of n: Then, using also the lower bound aj bke j and supposing that 4 e KXKðeÞ; it is easily seen that F ðxÞXJV K in a subinterval of length b1 of the interval ½n 1; n þ 1 : Hence SðK; JV 4 K e ÞbTðJÞðJV 4 K e Þ2 : Combined with (1.6), this inequality implies that TðJÞJV 12 5K 4þ4e J 2 ; and summing over dyadic values of J and V we obtain the estimate (1.7).
&
Acknowledgments The author is grateful to Professors A. Ivic´ and Y. Motohashi for valuable comments.
References [1] Roelof W. Bruggeman, Yoichi Motohashi, A new approach to the spectral theory of the fourth moment of the Riemann zeta-function, J. Reine Angew. Math., to appear. [2] D.R. Heath-Brown, The twelfth power moment of the Riemann zeta-function, Quart. J. Math. Oxford 29 (2) (1978) 443–462. [3] A. Ivic´, On sums of Hecke series in short intervals, J. The´orie des Nombres de Bordeaux 13 (2001) 554–568. [4] A. Ivic´, On the moments of Hecke series at central points, Funct. Approx. 30 (2002) 49–82. [5] H. Iwaniec, Small eigenvalues of Laplacian for G0 ðNÞ; Acta Arith. 56 (1990) 65–82.
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[6] M. Jutila, The fourth moment of central values of Hecke series, in: M. Jutila, T. Metsa¨nkyla¨, W. de Gruyter (Eds.), Number Theory, Proceedings of the Turku Symposium on Number Theory in Memory of Kustaa Inkeri, Springer, Berlin, 2001, pp. 167–177. [7] M. Jutila, Y. Motohashi, A note on the mean value of the zeta and L-functions XI, Proc. Japan Acad. 78 (Ser. A) (2002) 1–6. [8] N.V. Kuznetsov, Sums of Kloosterman sums and the eighth power moment of the Riemann zetafunction, Tata Inst. Fund. Res. Stud. Math. 12 (1989) 57–117. [9] N.V. Kuznetsov, The Hecke Series at the Center of the Critical Strip preprint, Dal’nauka, Vladivostok, 1999, 27 pp. (in Russian). [10] N.M. Lebedev, Special Functions and their Application, Dover Publications, New York, 1972. [11] Y. Motohashi, Spectral Theory of the Riemann Zeta-Function, Cambridge University Press, Cambridge, 1997. [12] Y. Motohashi, A functional equation for the spectral fourth moment of modular Hecke L-functions, in: D.R. Heath-Brown, B.Z. Moroz (Eds.), Proceedings of the session in analytic number theory and Diophantine equations (Bonn, January–June 2002), Bonner Math. Schriften Nr. 360, Math. Inst. Univ. Bonn, Bonn 2003, 19pp; also available in arXiv e-print: math. NT/0310105. [13] E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, 1948.
Journal of Number Theory 108 (2004) 157–168
http://www.elsevier.com/locate/jnt
The twelfth moment of central values of Hecke series Matti Jutila Department of Mathematics, University of Turku, FIN-20014 Turku, Finland Received 2 October 2003; revised 18 March 2004
Communicated by B. Conrey
Abstract Let Hj ðsÞ be the Hecke L-function attached to the Maass wave form for the jth eigenvalue 14 þ k2j of the hyperbolic Laplacian acting in the Hilbert space of automorphic functions for the full modular group. The following mean value estimate for the central values Hj ð12Þ is proved: X 1 5K 4þe : Hj12 2 kj pK r 2004 Elsevier Inc. All rights reserved. Keywords: Maass forms; Hecke series; Central values
1. Introduction This paper is related to [6], where the fourth moment of the central values Hj ð12Þ of the Hecke series Hj ðsÞ ¼
N X
tj ðnÞns
n¼1
Fax: +358-2-3336595. E-mail address: jutila@utu.fi.
0022-314X/$ - see front matter r 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2004.05.012
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over a short spectral interval was estimated. We adopt the notation of the monograph [11] of Motohashi; thus the coefficients tj ðnÞ are the Hecke eigenvalues of the Maass wave form attached to the jth eigenvalue lj ¼ 14 þ k2j of the hyperbolic Laplacian. The Fourier coefficients rj ðnÞ of the form in question are connected with the Hecke eigenvalues by the relation rj ðnÞ ¼ tj ðnÞrj ð1Þ; and we write aj ¼ jrj ð1Þj2 =coshðpkj Þ as usual. In this notation, the main result of [6] was X 1 aj Hj4 5K 4=3þe ; 2 1=3
ð1:1Þ
jkj KjpK
actually, as a slight improvement (see [4, Eq. (4.12)]), the factor K e here can be due to Iwaniec [5], this replaced by ðlog KÞ16 : Together with the estimate aj bke j implied the ‘‘subconvexity’’ bound 1 1=3þe Hj : ð1:2Þ 5kj 2 This had been proved previously by Ivic´ [3] as a corollary of his estimate X 3 1 aj H j 5K 1þe 2 jk Kjp1
ð1:3Þ
j
and the non-negativity of Hj ð12Þ: Now, in view of (1.1) and (1.3), sums of the mixed form X 4 1 aj H j F ðxÞ ¼ 2 xpk pxþ1 j
for x^K are of interest. An optimal estimate would be F ðxÞ5K 1þe in analogy with (1.3), whereas (1.1) implies the unconditional bound F ðxÞ5K 4=3þe : The statistics of the values of F ðxÞ will be studied in terms of the mean square Z 2K SðKÞ ¼ F 2 ðxÞ dx: ð1:4Þ K
Also, let SðK; UÞ denote the same integral restricted to values of x satisfying F ðxÞXU: Theorem 1. We have SðKÞ5K 3þe
ð1:5Þ
SðK; UÞ5K 4þe U 1 :
ð1:6Þ
and
As a corollary of (1.6), we get (see Section 4) an estimate for the twelfth moment of Hj ð12Þ:
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Theorem 2. We have X kj pK
Hj12
1 5K 4þe : 2
ð1:7Þ
This can be viewed as an analogue of the well-known estimate Z
T
0
12 1 z þ it dt5T 2þe 2
ð1:8Þ
of Heath-Brown [2] which in fact will enter in the course of our argument. Also, note that (1.7) implies (1.2) as an immediate corollary and gives thus a third proof for the latter. As a novelty compared with (1.1) and (1.3), Theorem 2 shows that exceptionally large values of Hj ð12Þ can occur only very rarely. The proof of Theorem 1 will be based on a functional equation for weighted spectral sums of products Hj ðz1 ÞHj ðz2 ÞHj ðz3 ÞHj ðz4 Þ; where the z-points lie on or close to the critical line. On the other side of that functional equation, a similar sum appears with another quadruple of points and another spectral weight function. Such a curious relation, analogous to transformations of exponential sums by Poisson’s or Voronoi’s sum formula, was first observed by Kuznetsov [8] though essential corrections, due to Motohashi [12], were required in the formulation and proof of his claim. However, if the ‘‘parity sign’’ ej ¼ 71 of the jth Maass form is inserted as a factor to the product, then the problem becomes essentially easier. On the other hand, if at least one of the numbers zj equals 12; then this modification is harmless, for it follows from the functional equation of Hj ðsÞ that Hj ð12Þ ¼ 0 if ej ¼ 1: This observation was utilized by Motohashi in the context of his identity for sums of the type N X j¼1
aj tj ð f Þhðkj ÞHj2
1 2
(see [11, Lemma 3.8]), and this identity was the starting point in the proofs of (1.1) and (1.3). In the case of the spectral sum Hj4 ð12Þ; the quadruple ð12; 12; 12; 12Þ remains invariant in the transformation process and only the weight function is changed. We encountered this phenomenon in [6] by an ad hoc argument. Actually we did not get a functional equation strictly in the above sense, for the role of the power Hj4 ð12Þ was played by the product of Hj2 ð12Þ with a finite sum representing approximately another factor like this. An analogous phenomenon occurred in [7], where a spectral sum over jHj ð12 þ itÞj4 was transformed into another sum involving products Hj3 ð12ÞHj ð12 2itÞ: Turning to the method of the present paper, the proof of (1.5) will start (as in [6]) from a short spectral sum of aj Hj4 ð12Þ with a Gaussian weight. If the relevant values of kj in the original sum lie near x; then the weight function in the transformed sum is
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an oscillating function of x: When this relation is squared and integrated over xA½K; 2K ; a spectral double sum appears on the transformed side, and only those terms lying near the diagonal will be significant. Then, we get integrals of the type SðKÞ on both sides of the relation, but the new spectral size parameter—playing the role of K—will be smaller than K; the result is the inequality (3.13). Iterating this procedure, we end up eventually with a sufficiently short integral which can be estimated trivially without essential loss of accuracy, and the claim (1.5) follows. For a proof of (1.6), we combine the transformation device with the estimate (1.5) already established. Though the above argument could be carried out on the basis of the method of [6], it is more convenient to refer to an approximate variant of the fundamental functional equation given in [9] or [4]; the latter paper contains corrections and simplifications to the presentation in the former one. Also, an approach to this functional equation is outlined by Motohashi [12] as an easier variant of his sophisticated argument therein, and the details can be readily supplied following his guidelines.
2. The basic functional equation The following lemma of Kuznetsov ([9, Lemma 4.7]) is an approximate and specialized version of the above-mentioned functional equation. Lemma. Let K be a large parameter, 0oao1=2; and K a pQpK 1=2 : Then for x^K we have ! ! Z 8 kj x 2 1 2 N jzð12 þ irÞ rx 2 exp exp þ dr Q 2 p 0 jzð1 þ 2irÞj2 Q jX1 Z 8 X 1 1 N jzð12 þ irÞj aj Hj4 hðr; xÞ dr ¼ xQP6 ðlog xÞ þ hðkj ; xÞ þ 2 p 0 jzð1 þ 2irÞj2 jX1
X
aj Hj4
þ OðQ2 log6 KÞ:
ð2:1Þ
Here P6 is a polynomial of degree 6, and the function hðr; xÞ can be represented, for 15r5x; by an asymptotic expansion in 1=r with leading term of the form h0 ðr; xÞ ¼ bðr; xÞQr1=2 expðQ2 Z2 ðr=xÞ=4Þ sinðrZðx=rÞ þ xZðr=xÞ þ p=4Þ;
ð2:2Þ
where ZðyÞ ¼ 2 arsinh y and bðr; xÞ is a smooth stationary bounded function, i.e. @ i @ j bðr; xÞ 5i;j ri xj for i; jX0: @ri @x j
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Continuing the comments made in the introduction, let us further discuss the nature of the above functional equation and its proof. As we pointed out already, a somewhat less explicit transformation formula of the same flavor follows from the calculations in Section 3 of [6]. Indeed, in that paper, the weight function on the transformed side is essentially a factor of the order QK 1=2 times the real part of Z pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G2 ð12 þ irÞ22ir N bðyÞyir ð y1 þ 1 þ y1 Þ2ix Gð1 þ 2irÞ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 þ 1 þ y1 Þ2ir expðQ2 log2 ð y1 þ 1 þ y1 ÞÞ dy;
ð2:3Þ
where bðyÞ is stationary function of order y5=4 : The saddle point of the y-integral is ðx=rÞ2 ; and the saddle-point method yields a weight function of the form (2.2). Turning to a sketch of the proof of the lemma, one may start from the sum N X
ej aj Hj ðz1 ÞHj ðz2 ÞHj ðz3 ÞHj ðz4 Þ:
ð2:4Þ
j¼1
To begin with, letting the numbers zi lie in the domain of absolute convergence, this sum is expressed in terms of Kloosterman sums (up to some less significant terms) by use of the Spectral-Kloosterman sum formula of Bruggeman and Kuznetsov, next the resulting arithmetic sum is transformed by the functional equation of the Estermann zeta-function to another sum involving again Kloosterman sums, and finally the latter expression is translated back into the language of spectral theory by Kuznetsov’s Kloosterman-Spectral sum formula. The spectral expression for the sum (2.4) allows its analytic continuation to the central point with all zi ¼ 12; where the factors ej become irrelevant. This argument is indicated in [12], where the main focus, however, is the more difficult sum without the parity signs. The underlying reason for the difference between these two cases (i.e. with or without the parity signs) is that these lead, respectively, to the ‘‘opposite-sign’’ or ‘‘equal-sign’’ case of the Spectral-Kloosterman sum formula (see Theorems 2.2 and 2.4 in [11], and Eqs. (2.15) and (2.17) in [12]). In the former case, this formula involves the integral transform Z N Gðs þ irÞ rhðrÞ dr; ð2:5Þ N Gð1 s þ irÞ cos ps while in the latter case the corresponding transform is Z N Gðs þ irÞ rhðrÞ dr: N Gð1 s þ irÞ cosh pr
ð2:6Þ
An apparent advantage of the transform (2.5) over (2.6) is its exponential decay as a function of Im s:
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The above argument leads to a transformation formula with a certain weight function attached to Hj4 ð12Þ on the transformed side. The dominating part of the weight function is Z 1 4 1 s coshðprÞ sin2 ðpsÞjðsÞ ds; Gðs þ irÞGðs irÞG ð2:7Þ 2p5 i ðdÞ 2 where d is a small positive number and Z N Gðs iuÞGðs þ iuÞhðuÞu sinhðpuÞ du jðsÞ ¼ N
with ( ! !) ðu2 þ 1=4Þðu2 þ 9=4Þ ux 2 uþx 2 hðuÞ ¼ 2 exp þ exp : ðu þ 1=4Þðu2 þ 9=4Þ þ 626 Q Q To analyze the integral (2.7), one may use the formula (see [11, pp. 119–120]) Z 1 G2 ð12 sÞGðs þ irÞGðs irÞ sinðpsÞxs ds 2pi ðdÞ ( G2 ð12 þ irÞ 1 p F ð þ ir; 12 þ ir; 1 þ 2ir; 1=xÞxir ¼ 2 coshðprÞ Gð1 þ 2irÞ 2 ) G2 ð12 irÞ 1 ir F ð ir; 12 ir; 1 2ir; 1=xÞx : þ ð2:8Þ Gð1 2irÞ 2 We apply now the following property of Mellin transforms (which is a corollary of Theorem 73, that is Parseval’s formula, in [13]): if the Mellin transforms of the functions f ðxÞ and gðxÞ are F ðsÞ and GðsÞ; respectively, and the functions f ðxÞxc1=2 ; gðxÞxc1=2 lie in the Hilbert space L2 ð0; NÞ; then Z Z N 1 F ðsÞGðsÞ ds ¼ f ð1=xÞgðxÞx1 dx: 2pi ðcÞ 0 Writing the right-hand side of (2.8) as f ðx; rÞ for short, we see that the Mellin transform of f ð1=x; rÞ is 1 s Gðs þ irÞGðs irÞ sinðpsÞ; G2 2 and we may choose c ¼ d as in (2.7). We apply next the above formula to the pair f ðx; rÞ; f ðx; uÞ to write the integral (2.7) in terms of hypergeometric functions. Then, if the hypergeometric functions are transformed by the formula (see [10,
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Eq. (9.6.12)]) F ða; b; 2b; zÞ ¼
1þ
pffiffiffiffiffiffiffiffiffiffiffi!2a 0 pffiffiffiffiffiffiffiffiffiffiffi!2 1 1z 1 1 1 1z pffiffiffiffiffiffiffiffiffiffiffi A F @a; a b þ ; b þ ; 2 2 2 1þ 1z
and the u-integral is calculated, we end up with an integral of the type (2.3), and as we pointed out above, this yields the formula (2.2). In the course of these calculations, the new hypergeometric function can be expressed simply by the hypergeometric series if it converges rapidly, or else by the Gaussian integral. We remarked in the introduction that functional equations such as (2.1) or its generalizations can be viewed as analogues of the ‘‘Process B’’ in van der Corput’s method, that is transforming exponential sums by Poisson’s summation formula, and in view of the preceding discussion the analogy is indeed not only formal. The role of the Poisson summation is now played by the functional equation for Estermann’s functional equation, equivalent to the Voronoi summation. The original spectral sum is first transformed into an arithmetic form as a preparation for the ‘‘reflection’’ step—the core of the argument—after which the new arithmetic expression is again interpreted in terms of spectral theory. A methodical question naturally arising in the context of the proof of the functional equation is the curious role of the Kloosterman sums: why do these first appear and then disappear, without leaving any trace; does it mean that they are redundant, after all? Indeed, it was pointed out to the author by Prof. Motohashi that there ought to be an alternative and more functional approach to the functional equation avoiding Kloosterman sums, and it is our aim to return jointly to this topic elsewhere. To this end, one should interpret the functions Hj ðsÞ as L-functions attached to irreducible automorphic representations of the group G ¼ SLð2; RÞ rather than to particular automorphic forms, and similarly all functions of interest should ‘‘live’’ in the group G rather than in the upper half-plane. An excellent example of a methodical recasting like this is the recent paper [1] by Bruggeman and Motohashi, where a new proof for Motohashi’s spectral formula for a weighted fourth moment of Riemann’s zeta-function is given dispensing with the spectral theory of sums of Kloosterman sums.
3. Proof of Theorem 1 We prove first the estimate (1.5), which will then be needed in the proof of (1.6). We decompose the integral (1.4) defining SðKÞ according to the size of F ðxÞ: The contribution of the values x with F ðxÞpK 1þd to SðKÞ is 5K 3þ2d : Here d is a small positive number which will be fixed during the following discussion, and e is another small positive number which can be chosen sufficiently small in relation to d:
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Next we consider, for UXK 1þd ; the contribution, say S ðK; UÞ; of the values x such that UpF ðxÞp2U:
ð3:1Þ
Q ¼ UK 1e log K
ð3:2Þ
Put
with 0oeod; and cover the interval ½K 1; 2K by intervals of the form IðPÞ ¼ ½P; P þ Q : Let N be the set of the integers n such that (3.1) holds for some xA½n; n þ 1 : We classify the intervals IðPÞ according to the cardinality of the set N -IðPÞ: For a parameter JX1; let RðJÞ be the number of those intervals IðPÞ for which this cardinality lies in the interval ½J; 2JÞ: Then X S ðK; UÞ5 JRðJÞU 2 ; ð3:3Þ J
where J runs over the dyadic numbers 2n ; n ¼ 0; 1; 2; y : The number RðJÞ is now estimated by use of (2.1), which we write as AðxÞ ¼ xQP6 ðlog xÞ þ BðxÞ þ CðxÞ þ OðQ2 log6 KÞ
ð3:4Þ
for short. If IðPÞ is one of the RðJÞ intervals selected above, then AðxÞbJU for xAIðPÞ; note that the second term on the left-hand side of (2.1) is non-negative. By (3.2), the quantity AðxÞ ðbUÞ dominates over the first and last term in (3.4), so that AðxÞ5jBðxÞj þ jCðxÞj: Moreover, since hðr; xÞ5Qr1=2 ; we may omit the terms with kj pK e in BðxÞ: Let ˜ BðxÞ denote the remaining sum, where the leading term of the asymptotic expansion for hðkj ; xÞ is now given by the formula (2.2). We square the above-mentioned ˜ inequality, with BðxÞ replaced by BðxÞ; and integrate over the RðJÞ intervals IðPÞ: For an upper estimation of the integral arising on the right, we introduce a weight function wðxÞ; which is a smoothed version of the characteristic function of ½K; 2K : Then Z Z 2 ˜ RðJÞQðJUÞ2 5 jBðxÞj wðxÞ dx þ jCðxÞj2 wðxÞ dx: ð3:5Þ Consider next the integral Z X 2 ˜ wðxÞ dx ¼ jBðxÞj kj ;kk 4K e
aj ak Hj4
Z 1 1 Hk4 wðxÞhðkj ; xÞhðkk ; xÞ dx: 2 2
ð3:6Þ
Here, we replace h by the leading term h0 noting that the other terms give analogous but smaller contributions. The exponential factor in (2.2) truncates the spectral sums
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in (3.6) to the following finite double sum: K e okj ; kk pk ¼ KQ1 log K ¼ K 2þe U 1 pK 1dþe pK 1d=2
ð3:7Þ
if we suppose that eod=2: The integrals on the right-hand side of (3.6) depend on the oscillatory nature of the integrand, that is on the sine factor, say sinðjðr; xÞÞ; in (2.2). Then jðr; xÞx ¼ arsinhðr=xÞ; and hence jjðr; xÞxr j^K 1 for the relevant values of the variables. Thus, the integrand on the right-hand side of (3.6) is oscillatory if jkj kk jb1; and the double sum can be truncated to jkj kk j5K e ; for otherwise repeated integration by parts shows that the integral is negligibly small. On the other hand, if kj A½r; r þ 1 and kk A½s; s þ 1 with jr sj5K e ; then the integral in (3.6) can be estimated by 5KQ2 r1 ð1 þ jr sjÞA
ð3:8Þ
for any fixed large number A: Since this depends only on r and s but not on kj and kk ; the summations over these pairs ðkj ; kk Þ in (3.6) can be separated in the upper estimation. Next note that X X 1 1 1 2 4 4 aj H j ak Hk p ðF ðrÞ þ F 2 ðsÞÞ: 2 2 2 spkk osþ1 rpkj orþ1 Multiplying this by the expression (3.8), integrating over ðr; sÞA½K e ; k ½K e ; k with s ¼ r þ OðK e Þ; and recalling the choice (3.2) of Q; we obtain Z
2 ˜ jBðxÞj wðxÞ dx5K 12e U 2 log2 K
Z
k
r1 F 2 ðrÞ dr;
1
the reappearance of the function F is an important phenomenon here. Hence Z
2 ˜ jBðxÞj wðxÞ dx5K 12e U 2 log3 K max L1 SðLÞ:
ð3:9Þ
Lpk
Similarly, noting that jzð1 þ 2irÞj1 5log r for rX2; we have Z
2
jCðxÞj wðxÞ dx5K
12e
2
6
Z
k
U log K 1
16 1 r z þ ir dr: 2 1
By (1.8), the standard estimate for the zeta-function, and (3.7), this is further 5K 1ð5=3Þe U 2 k5=3 ¼ K 7=3 U 1=3 :
ð3:10Þ
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Combining now (3.3), (3.5), (3.9), and (3.10), we get X J 1 RðJÞQðJUÞ2 S ðK; UÞ5 Q1 J
5 K U log2 K max L1 SðLÞ þ K 10=3þe U 2=3 : e
Lpk
ð3:11Þ
Turning to the estimation of SðKÞ; let U run over the numbers 2n K 1þd for n ¼ 0; 1; 2; y and note that X SðKÞ5K 3þ2d þ S ðK; UÞ: U
Thus SðKÞ5K 3þ2d þ S ðK; UÞ log K
ð3:12Þ
for some UXK 1þd : Since the last term on the right-hand side of (3.11) is dominated by the first term on the right-hand side of (3.12), we obtain SðKÞ5K 3þ2d þ K e U log3 K max L1 SðLÞ: Lpk
This implies that SðKÞ SðLÞ 51 þ max 3þ2d ðL2þ2d K 32de U log3 KÞ: Lpk L K 3þ2d By (3.7) and the assumption UXK 1þd ; the last factor on the right-hand side is 2
5K ð2þ2dÞð2þeÞ32de U 12d log3 K5K dþeþ2ed2d log3 K51: Thus, since kpK 1d=2 ; we conclude that SðKÞ SðLÞ 51 þ max 3þ2d 1d=2 K L3þ2d 1pLpK
ð3:13Þ
and (1.5) follows if this relation is iterated sufficiently many times and the quotient occurring at the last stage is estimated trivially. For a proof (1.6), we proceed as above with slight modifications. We let now e stand generally for a small positive number, not necessarily the same at each occurrence. For U5K 1þe ; the estimate (1.6) follows from (1.5). Otherwise K 1þe 5U5K 2þe by the ‘‘trivial’’ estimate F ðxÞ5x2þe ; and by (3.11), (1.5), and (3.7) we have S ðK; UÞ5 ðk2 U þ K 10=3 U 2=3 ÞK e 5 K 4þe U 1 ð1 þ U 1=3 K 2=3 Þ5K 4þe U 1 :
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This completes the proof of (1.6), since the same estimate follows for SðK; UÞ as well by summation over dyadic values of U: &
4. Proof of Theorem 2 The contribution of the terms with Hj ð12ÞoK 1=4 to the sum (1.7) is 5K 4þe in view of the known estimate for the fourth moment of Hj ð12Þ: We classify the remaining terms according to the condition V pHj ð12Þo2V ;
ð4:1Þ
where V XK 1=4 runs over dyadic values. For nA½K; 2KÞ; either the condition (4.1) holds for no spectral parameter kj A½n; n þ 1 ; or else there is a non-negative integer n such that this condition holds for at least 2n and less than 2nþ1 parameters. We now estimate the contribution of those terms corresponding to a certain value J ¼ 2n ; let TðJÞ be the number of the relevant values of n: Then, using also the lower bound aj bke j and supposing that 4 e KXKðeÞ; it is easily seen that F ðxÞXJV K in a subinterval of length b1 of the interval ½n 1; n þ 1 : Hence SðK; JV 4 K e ÞbTðJÞðJV 4 K e Þ2 : Combined with (1.6), this inequality implies that TðJÞJV 12 5K 4þ4e J 2 ; and summing over dyadic values of J and V we obtain the estimate (1.7).
&
Acknowledgments The author is grateful to Professors A. Ivic´ and Y. Motohashi for valuable comments.
References [1] Roelof W. Bruggeman, Yoichi Motohashi, A new approach to the spectral theory of the fourth moment of the Riemann zeta-function, J. Reine Angew. Math., to appear. [2] D.R. Heath-Brown, The twelfth power moment of the Riemann zeta-function, Quart. J. Math. Oxford 29 (2) (1978) 443–462. [3] A. Ivic´, On sums of Hecke series in short intervals, J. The´orie des Nombres de Bordeaux 13 (2001) 554–568. [4] A. Ivic´, On the moments of Hecke series at central points, Funct. Approx. 30 (2002) 49–82. [5] H. Iwaniec, Small eigenvalues of Laplacian for G0 ðNÞ; Acta Arith. 56 (1990) 65–82.
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[6] M. Jutila, The fourth moment of central values of Hecke series, in: M. Jutila, T. Metsa¨nkyla¨, W. de Gruyter (Eds.), Number Theory, Proceedings of the Turku Symposium on Number Theory in Memory of Kustaa Inkeri, Springer, Berlin, 2001, pp. 167–177. [7] M. Jutila, Y. Motohashi, A note on the mean value of the zeta and L-functions XI, Proc. Japan Acad. 78 (Ser. A) (2002) 1–6. [8] N.V. Kuznetsov, Sums of Kloosterman sums and the eighth power moment of the Riemann zetafunction, Tata Inst. Fund. Res. Stud. Math. 12 (1989) 57–117. [9] N.V. Kuznetsov, The Hecke Series at the Center of the Critical Strip preprint, Dal’nauka, Vladivostok, 1999, 27 pp. (in Russian). [10] N.M. Lebedev, Special Functions and their Application, Dover Publications, New York, 1972. [11] Y. Motohashi, Spectral Theory of the Riemann Zeta-Function, Cambridge University Press, Cambridge, 1997. [12] Y. Motohashi, A functional equation for the spectral fourth moment of modular Hecke L-functions, in: D.R. Heath-Brown, B.Z. Moroz (Eds.), Proceedings of the session in analytic number theory and Diophantine equations (Bonn, January–June 2002), Bonner Math. Schriften Nr. 360, Math. Inst. Univ. Bonn, Bonn 2003, 19pp; also available in arXiv e-print: math. NT/0310105. [13] E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, 1948.