Number variance of the zeros of the Epstein zeta functions
type: Item
instance of: scholarly article
title: Number variance of the zeros of the Epstein zeta functions
zbMATH Open document ID: 0819.11031
author: Akio Fujii
published in: Proceedings of the Japan Academy. Series A
publication date: 1994-12-19T00:00:00Z
review text: Let \(\zeta (s,Q)\) be the Epstein zeta-function of an integral form \(Q= ax^ 2+ bxy+ cy^ 2\), of discriminant \(d\), and let \(k= \sqrt{d} /2a\). It was shown by \textit{H. M. Stark} [Mathematika 14, 47-55 (1967; Zbl 0242.12010)] that the zeros of \(\zeta (s,Q)\) are very well behaved for \(| \text{Im}(s)| \leq 2k\). In particular if \(N(T)\) is the number of zeros with \(-1<\sigma <2\) and \(0\leq t\leq T\), then \(N(T)= {T\over \pi}\log ({{kT} \over {\pi e}})+ S(T)\), with \(S(T)= {1\over\pi} \arg \zeta({1\over 2}+ 2iT)+ O(1)\) for \(T\leq 2k\). The present paper shows that \(\int_ T^{2T-2} \{S(t+ \alpha)- S(t)\}^ 2 dt \ll T\) uniformly for \(T\leq k\) and \(\alpha\leq 1\). If the GUE law for the distribution of the zeros held for \(T\leq k\) the above integral would have been asymptotically \(CT\log (kT)\) for \(1\ll \alpha\leq 1\). In fact the paper proves an asymptotic formula for \[ \int_ T^{2T} \biggl( \arg {{\zeta (\sigma+ i(t+ h))} \over {\zeta (\sigma+ it)}} \biggr)^ 2 dt, \] uniformly for \(h\leq T\), for any fixed \(\sigma\in ({1\over 2}, 1]\).
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11E45
zbMATH DE Number: 701469
zbMATH Keywords: variance
zbMATH Keywords: binary quadratic form
zbMATH Keywords: Epstein zeta-function
zbMATH Keywords: number of zeros
zbMATH Keywords: GUE law
zbMATH Keywords: asymptotic formula
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
full work available at URL: pjaa.70.140
OpenAlex ID: W2011694903
cites work: Semiclassical formula for the number variance of the Riemann zeros
cites work: On Epstein's zeta function
cites work: Q3778823
cites work: On Indefinite Quadratic Forms in Five Variables
cites work: Q3137237
cites work: On the distribution of the zeros of the Riemann zeta function in short intervals
cites work: Q3809873
cites work: The Riemann zeta-function. Transl. from the Russian by Neal Koblitz
cites work: Hilbert's Inequality
cites work: Q3993105
cites work: On the zeros of Epstein's zeta function
DOI: 10.3792/PJAA.70.140
description: scientific article; zbMATH DE number 701469