An explicit mean-value estimate for the PNT in intervals

Abstract

This paper gives an explicit version of Selberg's 1943 mean-value estimate for the prime number theorem in intervals under the Riemann hypothesis. Two applications are given: for primes in short intervals, and Goldbach numbers (sums of two primes) in short intervals. Under the Riemann hypothesis, we show there exists a prime in $(y,y+32277{\log }^{2}y]$ for at least half the $y\in [x,2x]$ for all $x\ge 2$, and at least one Goldbach number in $(x,x+9696{\log }^{2}x]$ for all $x\ge 2$.

arXiv commentAccepted for publication in the Journal of the Australian Mathematical Society