On the distribution of zeros of linear combinations of Euler products
type: Item
instance of: scholarly article
title: On the distribution of zeros of linear combinations of Euler products
zbMATH Open document ID: 0853.11074
DOI: 10.1215/S0012-7094-95-08028-4
author: Dennis A. Hejhal
author: Enrico Bombieri
published in: Duke Mathematical Journal
publication date: 1997-01-09T00:00:00Z
review text: Let \(L_j (s)\) be functions defined by Dirichlet series, possessing Euler products, and all satisfying the same functional equation. The purpose of the present paper is to consider zeros of the linear combination \(f(s)= \sum^N_1 \lambda_j L_j (s)\). Let \(N(T)\) be the number of zeros of \(f\) up to height \(T\), and let \(N_0 (T)\) be the number of such zeros which are simple and lie on the critical line. Subject to suitable hypotheses it is then shown that both \(N(T)\) and \(N_0 (T)\) are asymptotic to \(CT \log T\), where \(C\) is a constant arising from the form of the \(\Gamma\)-factors in the functional equation. The proof is quite difficult, but its essence is to show that the behaviour of \(f(s)\) is usually dominated by just one of the functions \(L_j (s)\).
reviewed by: D. R. Heath-Brown
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 871846
zbMATH Keywords: simple zeros
zbMATH Keywords: Dirichlet series
zbMATH Keywords: Euler products
zbMATH Keywords: functional equation
MaRDI profile type: Publication
cites work: Q3794956
cites work: Q3778823
cites work: Q3820683
cites work: Simple Zeros of the Riemann Zeta-Function
cites work: Zeta functions of simple algebras
cites work: On the Zeros of Dirichlet L-Functions. I
cites work: Zeros on the critical line for Dirichlet series attached to certain cusp forms
cites work: Q3809873
cites work: Q3135212
cites work: Q4016337
cites work: Q4752704
cites work: On Euler Products and the Classification of Automorphic Forms II
cites work: Q4006621
cites work: Q3363327
cites work: Zeros of Hecke L-functions associated with cusp forms
cites work: Q5559720
cites work: Q3213195
cites work: Q4109725
cites work: Hilbert's Inequality
cites work: Q4143395
cites work: Analytic Proof of the Strong Multiplicity One Theorem
cites work: Zeros of principal \(L\)-functions and random matrix theory
cites work: Q5777299
cites work: Q3993105
cites work: Q5784349
cites work: Q5838486
cites work: Q5838487
cites work: Q5785897
cites work: Q3135199
cites work: Q5803461
cites work: Some Ω-theorems for the Riemann zeta-function
cites work: Q3882540
description: scientific article; zbMATH DE number 871846