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 $x\geq 2$, and at least one Goldbach number in $(x,x+9696 \log^2 x]$ for all $x\geq 2$.