On zeros of Hecke \(L\)-functions and their linear combinations on the critical line
type: Item
instance of: scholarly article
title: On zeros of Hecke \(L\)-functions and their linear combinations on the critical line
zbMATH Open document ID: 1233.11097
DOI: 10.1134/S1064562410020390
author: I. S. Rezvyakova
published in: Doklady Mathematics
publication date: 2010-11-17T00:00:00Z
review text: Let \(f(z)= \sum^\infty_{n=1} a(n)e(nz)\) be a holomorphic cusp form of weight \(k\), level \(D\) and character \(\chi\). Moreover, denote by \(L_f(s)=\sum^\infty_{n=1}a(n)n^{-s+(1-k)/2}\) the corresponding (normalized) \(L\)-function. Then \(L_f\) belongs to the Selberg class of degree 2. In particular, it satisfies a Riemann type functional equation with one gamma factor relating \(F(s)\) to \(\overline{F(1-\overline s)}\). With this notation the following result is formulated. Theorem 1: If \(f(z)\) is an eigenfunction of the Hecke algebra then a positive proportion of non-trivial zeros of \(L_f(s)\) lie on the critical line \(\sigma= 1/2\). In the second part of the paper the author considers the problem of zeros of linear combinations of Hecke \(L\)-functions, focusing again on these with real parts \(1/2\). The set up is as follows. Let \(\psi_j\) be complex characters of imaginary quadratic field \(\mathbb Q(\sqrt{-D})\), and let \(a_j\) be real numbers. Moreover, let \({\mathfrak f}(s)=\sum_j a_j L(s,\psi_j)\). Theorem 2: For a sufficiently large \(T\), the number of zeros \(\rho= (1/2)+ i\gamma\) of \({\mathfrak f}(s)\) with \(0<\gamma< T\) is \(\gg T(\log T)^{2/h} e^{-c\sqrt{\log\log T}}\), where \(h= h(-D)\) is the class number, and \(c\) is an absolute constant. Only very rough sketches of proofs are provided. One can judge that the author is building on classical ideas by Selberg and Hardy with further developments by Karatsuba, Voronin and Jutila. Details are promised to be given in forthcoming papers [Izv. Math. 74, No. 6, 1277--1314 (2010); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 74, No. 6, 183--222 (2010; Zbl 1233.11098) and Math. Notes 88, No. 3, 423--439 (2010); translation from Mat. Zametki 88, No. 3, 456--475 (2010; Zbl 1257.11083).
reviewed by: J. Kaczorowski
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 5816929
zbMATH Keywords: zeros on critical line
zbMATH Keywords: \(L\)-functions of cusp forms
zbMATH Keywords: Hecke \(L\)-functions
zbMATH Keywords: linear combinations of \(L\)-functions
MaRDI profile type: Publication
cites work: Zeros on the critical line for Dirichlet series attached to certain cusp forms
cites work: On the Zeros of Certain Dirichlet Series
cites work: Distribution of rational numbers in short intervals
full work available at URL: s1064562410020390
OpenAlex ID: W2039323399
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description: scientific article; zbMATH DE number 5816929