The Density of the Zeros of the Zeta Function and the Problem of the Distribution of Prime Numbers in Short Intervals

Abstract

It follows from the asymptotic formula for π(x) (Theorem 5 of Chapter VI) that there exists at least one prime number in every interval (x, x + y), where x > x 0>0 and $$ y = x\exp \left( { - c{{\left( {\frac{{\ln x}}{{\ln \ln x}}} \right)}^{0.6}}} \right). $$ An application of the Theorem on the density distribution of the zeros of the zeta function in the critical strip enables us to obtain a much stronger result (cf. the corollary of Theorem 2).