Simple bounds for the auxiliary function of Riemann

TL;DR

This paper provides simple, easy-to-compute numerical bounds for key functions related to the Riemann zeta function, aiding in their numerical evaluation.

Contribution

It offers straightforward bounds for functions like (s), (s), (s), and Z(t), serving as a practical reference for computational purposes.

Findings

Bounds are simple to compute and verify.

Useful for numerical calculations of (s) and Z(t).

Serves as a practical reference for researchers.

Abstract

We give simple numerical bounds for $\zeta(s)$, $\vartheta(s)$, $\mathop{\mathcal R}(s)$, $Z(t)$, for use in the numerical computation of these functions. The purpose of the paper is to give bounds for several functions needed in the calculation of $\zeta(s)$ and $Z(t)$. We are not pretending to have any originality. Our object is to be useful as a reference in our work for simple (simple to compute and simple to check) bounds.