On Goldbach numbers in arithmetic progressions (Q5939533)

Let \(G(q,k)\) denote the least even integer \(g\) with \(g\equiv k\) (mod \(q\)) that is written as the sum of two primes. In this paper, it is proved that for any fixed number \(J>14/13\), one has \(G(q,k)\ll q^J\) for all odd primes \(q\) and all integers \(k\), where the implied constant depends only on \(J\). This theorem improves the corresponding result of Jutila, in which the lower limit for \(J\) was \(6/5\) in place of \(14/13\). NEWLINENEWLINENEWLINEIn the proof, various techniques in the sieve theory are employed, and the sum \(\sum_{d|n, d<\sqrt{n}}\mu(d)\) plays a very important role, where \(\mu(d)\) denotes the Möbius function. In particular, it is the key observation that the latter sum vanishes whenever \(\mu(n)=1\). The proof also requires numerical estimates for several integrals, and the author evaluates them by hand. As he implies, however, if one would use a computer to estimate those integrals, then one could moderately reduce the limit for \(J\) in the final conclusion.