Note on the asymptotic of the auxiliary function

TL;DR

This paper refines the asymptotic approximation of the auxiliary function (s) to better understand its zero-free regions, building on Siegel's results and providing more precise asymptotic behavior as t approaches infinity.

Contribution

It introduces an improved approximation of (s) of the form f(s)(1+o(t)), enhancing the understanding of its asymptotic behavior and zero-free regions.

Findings

Derived an approximation of (s) as f(s)(1+o(t))

Clarified the asymptotic behavior of (s) for large t

Extended Siegel's results on (s) asymptotics

Abstract

To define an explicit regions without zeros of $\mathop{\mathcal R}(s)$, in a previous paper we obtained an approximation to $\mathop{\mathcal R}(s)$ of type $f(s)(1+U)$ with $|U|< 1$. But this $U$ do not tend to zero when $t\to+\infty$. In the present paper we get an approximation of the form $f(s)(1+o(t))$. We precise here Siegel's result, following his reasoning. This is essential to get the last Theorems in Siegel's paper about $\mathop{\mathcal R}(s)$.