Simple bounds for the auxiliary function of Riemann

We give simple numerical bounds for $\zeta (s)$, $\vartheta (s)$, $(s)$, $Z(t)$, for use in the numerical computation of these functions. The purpose of the paper is to give bounds for several functions needed in the calculation of $\zeta (s)$ and $Z(t)$. We are not pretending to have any originality. Our object is to be useful as a reference in our work for simple (simple to compute and simple to check) bounds.