High precision computation of Riemann’s zeta function by the Riemann-Siegel formula, I

We present rigorous and sharp bounds for the terms and remainder in the Riemann-Siegel formula (for a general argument, not necessarily on the critical line). This allows for the computation of ζ ( s ) ζ (s) and Z ( t ) Z(t) to high precision. We also derive the Riemann-Siegel formula in a new and more direct way.