Zeros of principal \(L\)-functions and random matrix theory
type: Item
instance of: scholarly article
title: Zeros of principal \(L\)-functions and random matrix theory
zbMATH Open document ID: 0866.11050
DOI: 10.1215/S0012-7094-96-08115-6
published in: Duke Mathematical Journal
publication date: 1997-06-30T00:00:00Z
review text: This paper studies the distribution of the distances between consecutive zeros of the Riemann zeta-function and of other primitive \(L\)-functions associated to cuspidal automorphic representations of \(\text{GL}_m\) over \({\mathbb Q}\). Write the zeros in the upper part of the critical strip as \({1\over2}+i\gamma_j\), ordered with respect to the real part of \(\gamma_j\), with multiplicities. The average distance between two consecutive numbers \(\tilde\gamma_j={m\over2\pi}\gamma_j\log |\gamma_j|\) is one; this follows from the asymptotic distribution of the \(\gamma_j\). The \(n\)-th level correlation between the first \(N\) of these normalized zeros is described by the quantity \[ R_n(B_N,f) = {n!\over N} \sum_{\{x_1, \ldots , x_n\}\subset B_N} f(x_1,\ldots, x_n), \] where \(B_N=\{\tilde\gamma_j : 1\leq j\leq N\}\), and where \(f\) runs through test functions on \(\mathbb R^n\) that are invariant under permutations of the coordinates and under translations in the direction \((1,1,\ldots,1)\). The asymptotic behavior of \(R_n(B_N,f)\) as \(N\rightarrow\infty\) is given under assumptions concerning the \(L\)-function under consideration and concerning \(f\). On the \(L\)-function one assumes the Riemann hypothesis and a mild technical condition (valid for \(m\leq 3\)). The test function \(f\) has to have rapid decay in all directions in the hyperplane \(H\) orthogonal to \((1,1,\ldots,1)\) and the support of its Fourier transform should be inside the region \(\sum_j |\xi_j|<2/m\). As \(N\rightarrow\infty\), the quantity \( R_n(B_N,f)\) tends to the integral \[ \sqrt n \int_H f(y) W_n(y)\,dy \] over the hyperplane \(H\). The density \(W_n(y)\) is equal to the determinant of the \(n\times n\)-matrix with \((i,j)\)-th element \(\sin\pi(y_i-y_j) /\pi(y_i-y_j)\). This density has also turned up in random matrix theory. This result is a consequence of the asymptotic behavior as \(T\rightarrow\infty\) of the more complicated quantity \[ R_n(T,f,h) = \mathop{\textstyle\s…
Mathematics Subject Classification ID: 11M50
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 908676
zbMATH Keywords: distances between consecutive zeros
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: primitive \(L\)-functions
zbMATH Keywords: \(n\)-th level correlation
zbMATH Keywords: zeros of \(L\)-functions
zbMATH Keywords: combinatorial sieving
zbMATH Keywords: correlation functions
zbMATH Keywords: explicit formula
zbMATH Keywords: random matrix theory
zbMATH Keywords: principal \(L\)-functions
Wikidata QID: Q56387049
author: Zeév Rudnick
author: Peter Sarnak
reviewed by: Roelof Bruggeman
MaRDI profile type: Publication
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full work available at URL: s0012-7094-96-08115-6
OpenAlex ID: W2068184834
description: scientific article; zbMATH DE number 908676