Exceptional zeros and the Goldbach problem
type: Item
instance of: scholarly article
title: Exceptional zeros and the Goldbach problem
zbMATH Open document ID: 1493.11140
author: Ade Irma Suriajaya
author: J. B. Friedlander
author: Daniel Alan Goldston
author: H. Iwaniec
published in: Journal of Number Theory
publication date: 2022-01-05T00:00:00Z
review text: The famous Goldbach conjecture predicts that every even integer exceeding two can be written as the sum of two primes. Following the normal practice in this subject, we find it simpler to consider a weighted sum over the representations involving the von Mangoldt function \(\Lambda(m)\). Let \[ G(n)=\sum_{\substack{m_1+m_2=n\\ 2 \nmid m_1m_2}}\Lambda(m_1)\Lambda(m_2). \] The Hardy-Littlewood conjecture predicts that, for \(n\) even, we have \(G(n)\sim \mathfrak{S}(n)n\), where \[ \mathfrak{S}(n)=2\prod_{p>2}\left( 1-\frac{1}{(p-1)^2} \right) \prod_{\substack{p|n \\ p>2}}\left(1+\frac{1}{(p-2)} \right).\] A rather weakened, but still formidable, form of the Hardy-Littlewood conjecture is as follows.\\ Weak Hardy-Littlewood-Goldbach Conjecture: For all sufficiently large even \(n\) we have \[ \delta\mathfrak{S}(n)n<G(n)<(2-\delta)\mathfrak{S}(n)n \] for some fixed \(0<\delta<1\). In this paper, the authors establish a zero-free region for any Dirichlet \(L\)-function under certain assumptions. The precise result is the following. Theorem 1. Assume that the Weak Hardy-Littlewood-Goldbach Conjecture holds for all sufficiently large even \(n\). Then there are no zeros of any Dirichlet \(L\)-function in the region \[ \sigma\geq1-\frac{c}{\log q(|t|+2)} \] with a positive constant \(c\) which is allowed to depend on \(\delta\).
reviewed by: Huafeng Liu
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11N35
zbMATH DE Number: 7452392
zbMATH Keywords: primes
zbMATH Keywords: zeros
zbMATH Keywords: Goldbach problem
Wikidata QID: Q114156909
MaRDI profile type: Publication
full work available at URL: j.jnt.2021.06.004
OpenAlex ID: W3186598588
cites work: GOLDBACH REPRESENTATIONS IN ARITHMETIC PROGRESSIONS AND ZEROS OF DIRICHLET L ‐FUNCTIONS
cites work: Q4043139
cites work: An application of the Hardy-Littlewood conjecture
cites work: On the conditional bounds for Siegel zeros
DOI: 10.1016/J.JNT.2021.06.004
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description: scientific article; zbMATH DE number 7452392