On accuracy of mathematical languages used to deal with the Riemann zeta function and the Dirichlet eta function

TL;DR

This paper introduces a novel computational approach using a new mathematical language that handles infinite and infinitesimal quantities to analyze the Riemann zeta and Dirichlet eta functions, aiming to improve numerical evaluation accuracy related to the Riemann Hypothesis.

Contribution

It proposes a new applied framework for infinite quantities, enabling more precise numerical computations of key functions related to the Riemann Hypothesis.

Findings

New approach allows evaluation of functions at specific points with improved accuracy

Comparison shows the new language enhances the description of infinite and infinitesimal quantities

Numerical examples demonstrate the effectiveness of the proposed method

Abstract

The Riemann Hypothesis has been of central interest to mathematicians for a long time and many unsuccessful attempts have been made to either prove or disprove it. Since the Riemann zeta function is defined as a sum of the infinite number of items, in this paper, we look at the Riemann Hypothesis using a new applied approach to infinity allowing one to easily execute numerical computations with various infinite and infinitesimal numbers in accordance with the principle `The part is less than the whole' observed in the physical world around us. The new approach allows one to work with functions and derivatives that can assume not only finite but also infinite and infinitesimal values and this possibility is used to study properties of the Riemann zeta function and the Dirichlet eta function. A new computational approach allowing one to evaluate these functions at certain points is proposed. Numerical examples are given. It is emphasized that different mathematical languages can be used to describe mathematical objects with different accuracies. The traditional and the new approaches are compared with respect to their application to the Riemann zeta function and the Dirichlet eta function. The accuracy of the obtained results is discussed in detail.