Application of the Argument Principle to Functions Expressed as Mellin Transforms

TL;DR

This paper presents a numerical method using the argument principle and Mellin transforms to count roots and poles of functions, demonstrated on the Riemann Zeta function, with potential for analytical estimates despite numerical inefficiency.

Contribution

It introduces a novel numerical algorithm leveraging Mellin transforms and convolutions to evaluate root counts of complex functions, including the Riemann Zeta function.

Findings

The method can determine the number of roots minus poles within a region.

Application to the Riemann Zeta function demonstrates the approach.

The procedure is numerically inefficient but may allow analytical estimates.

Abstract

We describe a numerical algorithm for evaluating the numbers of roots minus the number of poles contained in a region based on the argument principle with the function of interest being written as a Mellin transformation of a usually simpler function. Because the function to be transformed may be simpler than its Mellin transform whose roots are to be sought we express the final integrals in terms of the former accepting higher dimensional integrals. Nonlinear terms are expressed as convolutions approximating reciprocal values by exponential sums. As an example the final expression is applied to the Riemann Zeta function. The procedure is very inefficient numerically. However, depending on the function to be investigated it may be possible to find analytical estimates of the resulting integrals.