The numerical evaluation of the Riesz function

TL;DR

This paper develops a numerical method to evaluate the Riesz function for large x, analyzing specific cases related to the Riemann hypothesis, and observes decay patterns consistent with theoretical predictions.

Contribution

It introduces a numerical scheme for computing the Riesz function and visualizes its asymptotic behavior for key parameter cases linked to the Riemann hypothesis.

Findings

The Riesz function exhibits x^{-1/4} decay for m=2, p=1.

The Riesz function exhibits x^{-3/4} decay for m=p=2.

Oscillatory behavior is observed, supporting theoretical conjectures.

Abstract

The behaviour of the generalised Riesz function defined by \[S_{m,p}(x)=\sum_{k=0}^\infty \frac{(-)^{k-1}x^k}{k! \zeta(mk+p)}\qquad (m\geq 1,\ p\geq 1)\] is considered for large positive values of $x$. A numerical scheme is given to compute this function which enables the visualisation of its asymptotic form. The two cases $m=2$, $p=1$ and $m=p=2$ (introduced respectively by Hardy and Littlewood in 1918 and Riesz in 1915) are examined in detail. It is found on numerical evidence that these functions appear to exhibit the $x^{-1/4}$ and $x^{-3/4}$ decay, superimposed on an oscillatory structure, required for the truth of the Riemann hypothesis.