Barban-Davenport-Halberstam average sum and exceptional zero of \(L\)-functions
type: Item
instance of: scholarly article
title: Barban-Davenport-Halberstam average sum and exceptional zero of \(L\)-functions
zbMATH Open document ID: 1182.11041
author: B. E. Eshmatov
published in: Journal of Number Theory
publication date: 2008-12-05T00:00:00Z
review text: Set \[ S(Q, x) = \sum_{q\leq Q}\sum_{(a, q)=1}\Big(\frac{x}{\varphi(q)}-\underset{n\leq x}{\sum_{n\equiv a\pmod q}}\Big)^2. \] The Barban-Davenport-Halberstam-theorem states that \(S(Q, x)\ll Qx\log x\) holds true for \(x\log^{-A} x\leq Q\leq x\). Montgomery proved \(S(Q, x)\sim Qx\log x\) for \(Q\) in the same range. In the present work the author gives an asymptotic formula for \(S(Q,x)\) valid in the range \(xe^{-c\sqrt{\log x}}\leq Q\leq x\), which, unavoidably, involves Siegel's zero. If no Siegel zero exists, this formula becomes \(S(Q,x)=Qx\log x + O\big(Qx\log^{1/2} x(\log\log x)^5\big)\). For the proof one has to evaluate the sum \[ \sum_{r\leq x-1}\tau_Q(r)\sum_{1\leq n\leq x-r}\Lambda(n)\Lambda(n+r), \] where \(\tau_Q(r)\) denotes the number of divisors \(d\) of \(r\) which are less than \(Q\). In this article the author applies the circle method to do so. The minor arc estimate is straightforward, on the major arcs the inner sum is expressed via the explicit formula. Since the contribution of an exceptional zero has to be treated like a second main term, this yields rather long expressions. The contribution of non-exceptional zeros is bounded using zero density estimates, the contribution of the main term and the exceptional zero yield rather complicated sums which have to be evaluated explicitly. These computations form the technical main part of the present article.
reviewed by: Jan-Christoph Schlage-Puchta
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11P55
zbMATH DE Number: 5378841
zbMATH Keywords: Barban-Davenport-Halberstam theorem
zbMATH Keywords: circle method
zbMATH Keywords: primes in arithmetic progressions
zbMATH Keywords: Siegel zero
zbMATH Keywords: large sieve
zbMATH Keywords: zero-density estimates for Dirichlet \(L\)-functions
MaRDI profile type: Publication
full work available at URL: j.jnt.2007.08.003
OpenAlex ID: W1987494994
cites work: THE 'LARGE SIEVE' METHOD AND ITS APPLICATIONS IN THE THEORY OF NUMBERS
cites work: Primes in arithmetic progressions
cites work: Q3900124
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cites work: On the Barban-Davenport-Halberstam Theorem-II
cites work: On the Barban-Davenport-Halberstam theorem: XIII
DOI: 10.1016/J.JNT.2007.08.003
description: scientific article; zbMATH DE number 5378841