Trivial zeros of Riemann auxiliary function

TL;DR

This paper proves that the Riemann auxiliary function has simple zeros at negative even integers, confirming a specific zero pattern predicted by earlier work.

Contribution

It establishes the simplicity of zeros at negative even integers for Siegel's Riemann auxiliary function, advancing understanding of its zero distribution.

Findings

$s=-2n$ are simple zeros of $ ext{Re}(s)$ for all $n extgreater 0$

Confirms zero pattern predicted by Siegel

Enhances knowledge of the zero structure of the Riemann auxiliary function

Abstract

It is proved that $s=-2n$ is a simple zero of $\mathop{\mathcal R}(s)$ for each integer $n\ge1$. Here $\mathop{\mathcal R}(s)$ is the function found by Siegel in Riemann's posthumous papers.