A Note on Primes and Goldbach Numbers in Short Intervals

Abstract

Let J(N, H) be the Selberg integral and E(x, T) the error term in Kaczorowski-Perelli's weighted form of the classical explicit formula. We prove that the estimate J(N, H) = o(H2 N) is connected with an appropriate estimate of ∫N2N| E(x,T)2dx, uniformly for H and T in some ranges. Moreover, assuming a suitable bound for ∫N2N| E(x,T)|2dx, we also obtain, for all sufficiently large N and H ≫ (log N)11/12, that every interval [N,N + H] contains ≫ H Goldbach numbers.