A note on the mean value of the zeta and \(L\)-functions. XI
type: Item
instance of: scholarly article
title: A note on the mean value of the zeta and \(L\)-functions. XI
zbMATH Open document ID: 0998.11025
author: Matti Jutila
author: Yoichi Motohashi
published in: Proceedings of the Japan Academy. Series A
publication date: 2002-08-18T00:00:00Z
review text: Let, in standard notation, \(H_j(s) = \sum_{n\geq 1}t_j(n)n^{-s} (\Re s > 1)\) be the Hecke series attached to the Maass wave form \(\psi_j(z)\) corresponding to the eigenvalue \({1\over 4} + \kappa_j^2\) of the non-Euclidean Laplacian. The reviewer [J. Théor. Nombres Bordx. 13, 453-468 (2001; Zbl 0994.11020)] has shown that \[ \sum_{K\leq \kappa_j \leq K+1}\alpha_j H_j^3(\textstyle{1\over 2}) \ll K^{1+\varepsilon} \] with \(\alpha_j = |\rho_j(1)|^2/\text{ {cosh}}(\pi\kappa_j)\), where \(\rho_j(1)\) is the first Fourier coefficient of \(\psi_j(z)\). The proof depended on \textit{Y. Motohashi}'s explicit formula [see his tract, `Spectral theory of the Riemann zeta-function', Cambridge (1997; Zbl 0887.11001)] for the sum \(\sum_{j\geq 1}\alpha_j H_j^2(\textstyle{1\over 2})t_j(f)h(\kappa_j)\), where \(h\) is a suitable even function of fast decay. Using a similar approach \textit{M. Jutila} [Number Theory, in Memory of K. Inkeri, de Gruyter, Berlin, 167-177 (2001; Zbl 0972.11041)] proved that \[ \sum_{K\leq \kappa_j \leq K+K^{1/3}}\alpha_j H_j^4(\textstyle{1\over 2}) \ll K^{{4\over 3}+\varepsilon}.\leqno(1) \] In the present paper the authors generalize (1) by proving that \[ \sum_{K\leq \kappa_j \leq K+K^{1/3}}\alpha_j |H_j^4(\textstyle{1\over 2}+it)|\ll K^{{4\over 3}+\varepsilon}\leqno(2) \] for \(0 \leq t \leq K^{1-\vartheta}, {1\over 3} < \vartheta < 1\), where the \(\ll\)-constant depends only on \(\varepsilon\) and \(\vartheta\). To prove (2), the authors first derive an approximate functional equation for \(H_j^2({1\over 2})\). The problem then becomes the estimation of \[ \sum_{m,n\geq 1}\psi(m)\psi(n)d(m)d(n)(mn)^{-1/2}(m/n)^{it} \sum_{j\geq 1}\alpha_jh(\kappa_j)t_j(m)t_j(n), \] where \(\psi\) is a suitable test-function and \(d(n)\) is the number of divisors of \(n\). The sum over \(j\) is transformed by the Kuznetsov trace formula (Motohashi, op. cit., Theorem 2.2). The ensuing sum over \(n\) is transformed by the Voronoi summation formula. After this, the p…
reviewed by: Aleksandar Ivić
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11F37
zbMATH DE Number: 1782977
zbMATH Keywords: hybrid subconvexity bound
zbMATH Keywords: Hecke \(L\)-function
zbMATH Keywords: Maass form
zbMATH Keywords: Bruggeman-Kuznetsov sum formula
zbMATH Keywords: binary additive divisor sum
zbMATH Keywords: saddle point method
zbMATH Keywords: spectral large sieve
zbMATH Keywords: spectral fourth moment
MaRDI profile type: Publication
full work available at URL: pjaa.78.1
OpenAlex ID: W1973974554
cites work: On sums of Hecke series in short intervals
cites work: The spectral growth of automorphic L-functions.
cites work: Lectures on a method in the theory of exponential sums
cites work: The fourth moment of central values of Hecke series
cites work: Q5202254
cites work: The binary additive divisor problem
cites work: Q4357058
cites work: A note on the mean value of the zeta and \(L\)-functions. IX
cites work: Q5846799
DOI: 10.3792/PJAA.78.1
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description: scientific article; zbMATH DE number 1782977