An explicit mean-value estimate for the PNT in intervals

TL;DR

This paper provides an explicit version of Selberg's mean-value estimate for the prime number theorem under the Riemann hypothesis, with applications to primes and Goldbach numbers in short intervals.

Contribution

It offers a new explicit bound for primes and Goldbach numbers in short intervals assuming the Riemann hypothesis, improving previous estimates.

Findings

Existence of a prime in $(y,y+32277\log^2 y]$ for half of $y\in[x,2x]$

Existence of a Goldbach number in $(x,x+9696\log^2 x]$ for all $x\geq 2$

Explicit bounds under the Riemann hypothesis for primes and Goldbach numbers in short 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 $x\geq 2$.