A note on the number of primes in short intervals

Abstract:

Let $J\left ( {\beta ,T} \right ) = \int _1^{{T^\beta }} {{{\left ( {\sum \nolimits _{x < {p^k} \leq x + x/T} {\log p - x/T} } \right )}^2}dx/{x^2}}$, where the sum is over prime powers. H. L. Montgomery has shown that on the Riemann hypothesis, there is a positive constant ${C_0}$ such that for each $\beta \geq 1,J\left ( {\beta ,T} \right ) \leq {C_0}\beta {\log ^2}T/T$, provided that $T$ is sufficiently large. Here we prove a slightly stronger result from which we deduce a lower bound of the same order.