Bounding the integral of the argument of the Riemann Zeta function

TL;DR

This paper improves bounds on the integral of the argument of the Riemann zeta function, aiding in the precise counting of zeta zeros using Turing's method.

Contribution

It provides a tighter estimate of the integral of the argument of the Riemann zeta function, enhancing methods for zero counting.

Findings

Improved bounds on the integral of the argument of zeta

Enhanced accuracy in zero counting methods

Refined estimates for Turing's method

Abstract

This article improves the estimate of $|S_1(t_2)-S_1(t_1)|$, which is the definite integral of the argument of the Riemann zeta-function between $t_1$ and $t_2$. Estimates of this quantity are needed to apply Turing's method to compute the exact number of zeta zeros up to a given height.