Asymptotic Expansions of the auxiliary function

TL;DR

This paper provides explicit asymptotic expansions of the auxiliary function related to the Riemann-Siegel formula, extending their validity, identifying zero-free regions, and analyzing its behavior for large imaginary parts.

Contribution

It explicitly derives and extends asymptotic developments of the auxiliary function and clarifies its zero-free regions, building on Siegel's earlier work.

Findings

Extended the range of validity of asymptotic expansions.

Identified regions where the function has no zeros.

Analyzed the asymptotic behavior of the function for large t.

Abstract

Siegel in 1932 published a paper on Riemann's posthumous writings, including a study of the Riemann-Siegel formula. In this paper we explicitly give the asymptotic developments of $\mathop{\mathcal R }(s)$ suggested by Siegel. We extend the range of validity of these asymptotic developments. As a consequence we specify a region in which the function $\mathop{\mathcal R }(s)$ has no zeros. We also give complete proofs of some of Siegel's assertions. We also include a theorem on the asymptotic behaviour of $\mathop{\mathcal R }(\frac12-it)$ for $t \to+\infty$. Although the real part of $e^{-i\vartheta(t)}\mathop{\mathcal R }(\frac12-it)$ is $Z(t)$ the imaginary part grows exponentially, this is why for the study of the zeros of $Z(t)$ it is preferable to consider $\mathop{\mathcal R }(\frac12+it)$ for $t>0$.