Improvements to Turing's Method II

TL;DR

This paper presents improved bounds on the integral of the argument of the Riemann zeta-function, refining Turing's method for more accurate zero counting.

Contribution

It introduces tighter bounds on the integral of S(t), enhancing the effectiveness of Turing's method in analytic number theory.

Findings

New explicit bounds on the integral of S(t)

Enhanced accuracy in zero counting of the Riemann zeta-function

Refinement of Turing's method for better analytical estimates

Abstract

Turing's method uses explicit bounds on $|\int_{t_{1}}^{t_{2}} S(t)\, dt|$, where $\pi S(t)$ is the argument of the Riemann zeta-function. This article improves the bound on $|\int_{t_{1}}^{t_{2}} S(t)\, dt|$ given in an earlier paper by the author.