An explicit Selberg mean-value result with applications

Abstract

This paper gives an explicit version of Selberg's mean-value estimate under the Riemann hypothesis (RH) [20]. Two applications are given in short-interval results: one for primes, and one for Goldbach numbers. Under RH and for all x ≥ 2, we find at least one prime in (y, y + 37 log 2 y] for at least half the y ∈ [x, 2x], and at least one Goldbach number in (x, x + 864 log 2 x].

Explicit Estimates for the Distribution of Primes

“…This estimate is used to update two short-interval results: we prove that there are primes between cubes, that is, in intervals (n 3 , (n + 1) 3 ) for all n ≥ exp(exp(32.537)), and primes between n 155 and (n + 1) 155 for all n ≥ 1. These results are published in [1].…”

“…Under the Riemann hypothesis (RH), we prove an explicit error estimate and an explicit mean-value estimate for the PNT in short intervals. The former is published in [2] and the latter is in the preprint [3]. The mean-value estimate is based on Selberg's work [9], and is of particular interest for its applications, of which two are given.…”