On Euler products and multi-variate Gaussians.
type: Item
instance of: scholarly article
title: On Euler products and multi-variate Gaussians.
zbMATH Open document ID: 1048.11072
DOI: 10.1016/S1631-073X(03)00344-3
author: Dennis A. Hejhal
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-09-26T00:00:00Z
review text: Let \(F(s)=\sum\limits_n a_n n^{-s}\) be a Dirichlet series which converges absolutely for \(\Re s > 1\) and satisfies a certain functional equation. Under assumptions similar to those satisfied by the classical \(L\)-functions, \textit{A. Selberg} [Collected papers, Vol. 2 \S 44 (Berlin: Springer-Verlag) (1991; Zbl 0729.11001)] proved that when \(\dfrac 12 \leq \sigma \leq \dfrac 12 + (\log T)^{-\delta}\) for some \(\delta > 0\), both the real and imaginary parts of \(\kappa (\sigma , t)= (\pi \sum\limits_{p< t} | a_p| ^2 p^{-2\sigma})^{-1/2} \log F(\sigma + it)\) have a normal Gaussian distribution. In the present paper, the author considers the joint distribution in higher dimension of the logarithms of \(J\) such functions \(F_1, F_2, \cdots , F_J\), where each \(F_j\) is, in addition, an Euler product. It is also assumed that each pair of \(\log F_j\) and \(\log F_k\) is independent, that is, \[ \sum\limits_{p\leq x} p^{-1} c_j (p) \overline c_k (p) = \delta_{jk} A_j \log\log X + c_{ij} + O ((\log X)^{-v}) \] for certain positive constants \(A_j\) and \(c_{jk} \in \mathbb C\). The author states in detail the main results and gives an outline of its proof, which goes along the same line as Selberg's original argument.
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11M99
zbMATH DE Number: 1986152
zbMATH Keywords: value distribution
zbMATH Keywords: Euler products
reviewed by: Kai-Man Tsang
MaRDI profile type: Publication
cites work: Q3794956
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description: scientific article; zbMATH DE number 1986152