Simple bounds for the auxiliary function of Riemann

We give simple numerical bounds for ζ(s), ϑ(s), R(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 ζ(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.