Note on the asymptotic of the auxiliary function

To define an explicit regions without zeros of $(s)$, in a previous paper we obtained an approximation to $(s)$ of type $f(s)(1+U)$ with $\mathrm{\mid }U\mathrm{\mid }<1$. But this $U$ do not tend to zero when $t\to +\mathrm{\infty }$. 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 $(s)$.