Note on the asymptotic of the auxiliary function

Abstract

To define an explicit regions without zeros of $\mathcal{R}(s)$, in a previous paper we obtained an approximation to $\mathcal{R}(s)$ 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 $\mathcal{R}(s)$.

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