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

Anatolij A. Karatsuba and Melvyn B. Nathanson
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Anatolij A. Karatsuba: Steklov Mathematical Institute
Melvyn B. Nathanson: School of Mathematics, Institute for Advanced Study

Chapter Chapter VII in Basic Analytic Number Theory, 1993, pp 94-101 from Springer

Abstract: 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).

Date: 1993
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DOI: 10.1007/978-3-642-58018-5_7

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