On the Gram's Law in the Theory of Riemann Zeta Function

TL;DR

This paper investigates the distribution of zeros of the Riemann zeta function, focusing on Gram's law, and provides proofs for several of Selberg's formulas related to these zeros.

Contribution

It offers new proofs of Selberg's formulas and establishes statements about the zeros' distribution connected to Gram's law.

Findings

Proofs of Selberg's formulas on zeta zeros

Connections between Gram's law and zero distribution

Validation of certain assertions about zeros' distribution

Abstract

Some statements concerning the distribution of imaginary parts of zeros of the Riemann zeta\,-function are established. These assertions are connected with so\,-called `Gram's law' or `Gram's rule'. In particular, we give a proof of several Selberg's formulae stated him without proof in his paper `The Zeta Function and the Riemann Hypothesis' (1946), and some of their equivalents.