Sharper estimates for Chebyshev's functions $\vartheta$ and $\psi$

TL;DR

This paper improves estimates for Chebyshev's functions using recent zero-free regions and extensive zero calculations of the Riemann zeta function, refining classical bounds with modern computational data.

Contribution

It provides sharper bounds for Chebyshev's functions by leveraging new zero-free regions and extensive zero computations of the Riemann zeta function.

Findings

Improved bounds for Chebyshev's functions $ heta$ and $ ho$

Utilization of Kadiri's zero-free region

Incorporation of first $10^{13}$ zeros of the Riemann zeta function

Abstract

In this article we present some improved results for Chebyshev's functions $\vartheta$ and $\psi$ using the new zero-free region obtained by H. Kadiri and the calculated the first $10^{13}$ zeros of the Riemann zeta function on the critical line by Xavier Gourdon. The methods in the proofs are similar to those of Rosser-Shoenfeld papers on this subject.