THE LAPLACE TRANSFORM OF THE FOURTH MOMENT OF THE ZETA-FUNCTION. Aleksandar Ivić

Transcription

1 THE LAPLACE TRANSFORM OF THE FOURTH MOMENT OF THE ZETA-FUNCTION Aleksandar Ivić Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. (2), Abstract. The Laplace transform of ζ( 2 +ix) 4 is investigated, for which a precise expression is obtained, valid in a certain region in the complex plane. The method of proof is based on complex integration and spectral theory of the non-euclidean Laplacian.. INTRODUCTION Laplace transforms play an important rôle in analytic number theory. Of special interest in the theory of the Riemann zeta-function ζ(s) are the Laplace transforms (.) L k (s) := ζ( 2 + ix) 2k e sx dx (k N, Re s > ). E.C. Titchmarsh s well-known monograph [2, Chapter 7] gives a discussion of L k (s) when s = σ is real and σ +, especially detailed in the cases k = and k = 2. Indeed, a classical result of H. Kober [4] says that, as σ +, (.2) L (2σ) = γ log(4πσ) 2 sin σ N + c n σ n + O(σ N+ ) n= for any given integer N, where the c n s are effectively computable constants and γ = is Euler s constant. For complex values of s the function L (s) 99 Mathematics Subject Classification. M6, F72, F66. Key words and phrases. Riemann zeta-function, Laplace transform, complex integration, spectral theory. Typeset by AMS-TEX

2 2 Aleksandar Ivić was studied by F.V. Atkinson [], and more recently by M. Jutila [3], who noted that Atkinson s argument gives (.3) L (s) = ie 2 is (log(2π) γ + ( π 2 s)i) + 2πe 2 is n= d(n) exp( 2πine is ) + λ (s) in the strip < Re s < π, where the function λ (s) is holomorphic in the strip Re s < π. Moreover, in any strip Re s θ with < θ < π, we have λ (s) θ ( s + ). In [2] M. Jutila gave a discussion on the application of Laplace transforms to the evaluation of sums of coefficients of certain Dirichlet series. F.V. Atkinson [2] obtained the asymptotic formula (.4) L 2 (σ) = σ (A log4 σ + B log3 σ + C log2 σ + D log σ + E) + λ 2(σ), where σ +, and A = 2π 2, B = π 2 (2 log(2π) 6γ + 24ζ (2)π 2 ) (.5) λ 2 (σ) ε ( σ ) 3 4 +ε. He also indicated how, by the use of estimates for Kloosterman sums, one can improve the exponent 3 4 in (.5) to 8 9. This is of historical interest, since it is one of the first instances of application of Kloosterman sums to analytic number theory. Atkinson in fact showed that (σ = Re s > ) (.6) L 2 (s) = 4πe 2 s d 4 (n)k (4πi ne 2 s ) + φ(s), n= where d 4 (n) is the divisor function generated by ζ 4 (s), K is the Bessel function, and the series in (.6) as well as φ(s) are both analytic in the region s < π. When s = σ + one can use the asymptotic formula K (z) = 2 πz /2 e z ( 8z + O( z 2 ) ) ( arg z < θ < 3π, z ) 2

3 Laplace transform of the fourth moment of ζ( 2 + ix) 3 and then, by delicate analysis, one can deduce (.4) (.5) from (.6). The author [5] gave explicit, albeit complicated expressions for the remaining coefficients C, D and E in (.4). More importantly, he applied a result on the fourth moment of ζ( 2 + it), obtained jointly with Y. Motohashi [9], [] (see also [4]), to establish that (.7) λ 2 (σ) σ /2 (σ +), and this is where the matter presently rests. For k 3 not much is known about L k (s), even when s = σ +. This is not surprising, since not much is known about upper bounds for I k (T ) := T ζ( 2 + it) 2k dt (k 3, k N). For a discsussion on I k (T ) the reader is referred to the author s monographs [3] and [4]. One trivially has (.8) I k (T ) e ζ( 2 + it) 2k e t/t dt = el k ( T ). Thus any nontrivial bound of the form (.9) L k (σ) ε ( σ ) ck +ε (σ +, c k ) gives, in view of (.8) (σ = /T ), the bound (.) I k (T ) ε T c k+ε. Conversely, if (.) holds, then we obtain (.9) from the identity L k ( T ) = T I k (t)e t/t dt, which is easily established by integration by parts.

4 4 Aleksandar Ivić 2. SPECTRAL THEORY AND THE LAPLACE TRANSFORM OF ζ( 2 + ix) 4 The purpose of this paper is to consider L 2 (s), where s is a complex variable, and to prove a result analogous to (.3), valid in a certain region in C. We shall not use Atkinson s method and try to elaborate on (.6). Our main tools are powerful methods from spectral theory, by which recently much advance has been made in connection with I 2 (T ). For a competent and extensive account of spectral theory the reader is referred to Y. Motohashi s monograph [9]. Some of the relevant papers on I 2 (T ) are [6] [], [4] [8] and [2]. We begin by stating briefly the necessary notation involving the spectral theory of the non-euclidean Laplacian. As usual {λ j = κ 2 j + 4 } {} will denote the discrete spectrum of the non-euclidean Laplacian acting on SL(2, Z) automorphic forms, and α j = ρ j () 2 (cosh πκ j ), where ρ j () is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue λ j to which the Hecke series H j (s) is attached. We note that (2.) κ j K Our result is the following α j H 3 j ( 2 ) K2 log C K (C > ). THEOREM. Let φ < π 2 have be given. Then for < s and arg s φ we (2.2) L 2 (s) = s (A log4 s + B log3 s + C log2 s + D log s + E) + G 2(s) + s 2 α j Hj 3 ( 2 ) ( s iκ j R(κ j )Γ( 2 + iκ j) + s iκ j R( κ j )Γ( 2 iκ j) ), j= where (2.3) R(y) := π (2 iy Γ( 4 i 2 y) )3 2 Γ( 4 + i Γ(2iy) cosh(πy) 2y) and in the above region G 2 (s) is a regular function satisfying (C > is a suitable constant) (2.4) { G 2 (s) s /2 C log( s } + 2) exp. (log log( s + 2)) 2/3 (log log log( s + 2)) /3

5 Laplace transform of the fourth moment of ζ( 2 + ix) 5 Remark. The constants A, B, C, D, E in (2.2) are the same ones as in (.4). Remark 2. From Stirling s formula for the gamma-function it follows that R(κ j ) κ /2 j. In view of (2.) this means that the series in (2.2) is absolutely convergent and uniformly bounded in s when s = σ is real. Therefore, when s = σ +, (2.2) gives a refinement of (.7). Remark 3. From (.4) and (.7) it transpires that λ(σ) is an error term when < σ <. For this reason we considered the values < s in (2.2), although one could treat the case s > as well. Remark 4. From (2.2) and elementary properties of the Laplace transform one can easily obtain the Laplace transform of E 2 (T ) := T ζ( 2 + it) 4 dt T P 4 (log T ), P 4 (x) = 4 a j x j, where a 4 = /(2π 2 ) (for the evaluation of the remaining a j s, see [5]). 3. PROOF OF THE THEOREM Note first that in the integral defining L 2 (s) it suffices to consider only the range of integration [, ], since the range [, ] trivially contributes. We start from the well-known integral (3.) e z = 2πi (c) Γ(s)z s ds (Re z >, c > ), j= and the function Z 2 (w) := ζ( 2 + it) 4 t w dt (Re w > ), introduced and studied by Y. Motohashi [7], [9]. Here as usual (σ) σ+it = lim. T σ it Y. Motohashi showed that Z 2 (s) has meromorphic continuation over C. In the half-plane σ = Re s > it has the following singularities: the pole s = of order 5, simple poles at s = 2 ± iκ j (κ j = λ j 4 ) and poles at s = 2ρ, where ρ denotes complex zeros of ζ(s). The residue of Z 2 (s) at s = 2 + iκ h equals R (κ h ) = π 2 (2 iκ Γ( h 4 i 2 κ ) h) 3 Γ(2iκ h ) cosh(πκ h ) Γ( 4 + i 2 κ h) κ j =κ h α j H 3 j ( 2 ),

6 6 Aleksandar Ivić and the residue at s = 2 iκ h equals R (κ h ). From (3.) we have, for c >, < s and arg s φ, (3.2) = ζ( 2 + ix) 4 e sx dx ( ) ζ( 2 + ix) 4 Γ(w)(sx) w dw dx 2πi (c) Γ(w)s w Z 2 (w) dw. = 2πi (c) In (3.2) we shift the line of integration to the contour L (w = u + iv) consisting of the curves (3.3) u = 2 C log 2/3 v (log log v ) /3 ( v v >, C > ) and the segment u = u, u = 2 C log 2/3 v (log log v ) /3, v v. Namely ζ(s) (see e.g., [3, Chapter 6]) for σ A(log t) 2/3 (log log t) /3 (s = σ + it, t t >, A > ) and a suitable constant A. The function Z 2 (w) will be regular on L, since the poles w = 2ρ, ζ(ρ) = ) lie to the left of L. For any given η > one has (3.4) Z 2 (w) e η Im w (w L) if C in (3.3) is taken sufficiently small. This follows easily from the proof of Lemma of [7], which shows that the order of Z 2 (w) is of the order given by (3.4) if Re w > and w stays away from the poles of Z 2 (w). In the forthcoming work [8] it is even shown that in the above region Z 2 (w) is of polynomial growth in v, which is more than what is required for our present purpose. Thus by the residue theorem we obtain from (3.2) (3.5) ζ( 2 + ix) 4 e sx dx = Res Γ(w)s w Z 2 (w) + Γ(w)s w Z 2 (w) dw, 2πi L and the last integral is regular for < s and arg s φ. There are residues at w = and at w = 2 ± iκ j. The contribution from w = is (see (.4)) s (A log4 s + B log3 s + C log2 s + D log s + E),

7 Laplace transform of the fourth moment of ζ( 2 + ix) 7 while the residues at w = 2 ± iκ j yield s /2 j= α j Hj 3 ( 2 ) ( s iκ j R(κ j )Γ( 2 + iκ j) + s iκ j R( κ j )Γ( 2 iκ j) ), where R(y) is defined by (2.3). Write Γ(w)s w Z 2 (w) dw = I + I 2, L say, where in I we have v V, and in I 2 we have v > V, where V ( ) is a parameter to be chosen. Since s w = s u e v arg s ( arg s φ), Γ(w) v u 2 e π 2 v (v ), then setting δ(x) = C log 2/3 x(log log x) /3 we obtain I s u + Similarly we have V v s ( 2 δ(v)) e ( π 2 φ)v dv s u + s ( 2 δ(v )). I 2 s 2 V e ( π 2 φ)v dv = s 2 e ( π 2 φ)v ( π 2 φ). Finally choosing V = C log( s + 2) with suitable C > we obtain (C 2 > ) Γ(w)s w Z 2 (w) dw L { s /2 C 2 log( s + 2) exp (log log( s + 2)) 2/3 (log log log( s + 2)) /3 which in view of (3.5) completes the proof of the Theorem. },

8 8 Aleksandar Ivić References [] F.V. Atkinson, The mean value of the zeta-function on the critical line, Quart. J. Math. Oxford (939), [2] F.V. Atkinson, The mean value of the zeta-function on the critical line, Proc. London Math. Soc. 47(94), [3] A. Ivić, The Riemann zeta-function, John Wiley and Sons, New York, 985. [4] A. Ivić, Mean values of the Riemann zeta-function, LN s 82, Tata Institute of Fundamental Research, Bombay, 99 (distr. by Springer Verlag, Berlin etc.). [5] A. Ivić, On the fourth moment of the Riemann zeta-function, Publs. Inst. Math. (Belgrade) 57(7)(995), -. [6] A. Ivić, The Mellin transform and the Riemann zeta-function, in Proceedings of the Conference on Elementary and Analytic Number Theory (Vienna, July 8-2, 996), Universität Wien & Universität für Bodenkultur, Eds. W.G. Nowak and J. Schoißengeier, Vienna 996, [7] A. Ivić, On the error term for the fourth moment of the Riemann zeta-function, J. London Math. Soc. 6(2)(999), [] A. Ivić, The Laplace transform of the fourth moment of the zeta-function, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. (2), [8] A. Ivić, M. Jutila and Y. Motohashi, The Mellin transform of power moments of the zetafunction, Acta Arithmetica 95(2), [9] A. Ivić and Y. Motohashi, A note on the mean value of the zeta and L-functions VII, Proc. Japan Acad. Ser. A 66(99), [] A. Ivić and Y. Motohashi, The mean square of the error term for the fourth moment of the zeta-function, Proc. London Math. Soc. (3)66(994), [] A. Ivić and Y. Motohashi, The fourth moment of the Riemann zeta-function, J. Number Theory 5(995), [2] M. Jutila, Mean values of Dirichlet series via Laplace transforms, in Analytic Number Theory (ed. Y. Motohashi), London Math. Soc. LNS 247, Cambridge University Press, Cambridge, 997, [3] M. Jutila, Atkinson s formula revisited, in Voronoi s Impact on Modern Science, Book, Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, 998, [4] H. Kober, Eine Mittelwertformel der Riemannschen Zetafunktion, Compositio Math. 3(936), [5] Y. Motohashi, The fourth power mean of the Riemann zeta-function, in Proceedings of the Amalfi Conference on Analytic Number Theory 989, eds. E. Bombieri et al., Università di Salerno, Salerno, 992, [6] Y. Motohashi, An explicit formula for the fourth power mean of the Riemann zeta-function, Acta Math. 7(993), [7] Y. Motohashi, A relation between the Riemann zeta-function and the hyperbolic Laplacian, Annali Scuola Norm. Sup. Pisa, Cl. Sci. IV ser. 22(995), [8] Y. Motohashi, The Riemann zeta-function and the non-euclidean Laplacian, Sugaku Expositions, AMS 8(995), [9] Y. Motohashi, Spectral theory of the Riemann zeta-function, Cambridge University Press, Cambridge, 997. [2] E.C. Titchmarsh, The theory of the Riemann zeta-function (2nd ed.), Clarendon Press, Oxford, 986. [2] N.I. Zavorotnyi, On the fourth moment of the Riemann zeta-function (in Russian), in Automorphic functions and number theory I, Coll. Sci. Works, Vladivostok, 989,

9 Laplace transform of the fourth moment of ζ( 2 + ix) 9 Aleksandar Ivić Katedra Matematike RGF-a Universitet u Beogradu, -Dušina 7 Beograd, Serbia (Yugoslavia) ivic@rgf.bg.ac.yu, aivic@matf.bg.ac.yu