An Explicit Mean-Value Estimate for the Prime Number Theorem in Intervals

Abstract This paper gives an explicit version of Selberg’s mean-value estimate for the prime number theorem in intervals, assuming the Riemann hypothesis 25. Two applications are given to short-interval results for primes and for Goldbach numbers. Under the Riemann hypothesis, we show there exists a prime in (y, y+32\, 277 ² y] for at least half the y x, 2x for all x 2, and at least one Goldbach number in (x, x+9696 ² x] for all x 2.