Exceptional zeros and the Goldbach problem
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 \(\delta\).
scientific article; zbMATH DE number 7452392
scientific article