1 1 Preliminaries 1 1.1 Function fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Primes and Divisors . . . . . . . . . . . . . . . . . . . . 2 1.2.2 The Picard Group . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Riemann-Roch . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Notation . . . . ...

In this article we derive a generalization of the Riemann-Siegel asymptotic formula for the Riemann zeta function. By subtracting the singularities closest to the critical point we obtain a significant reduction of the error term at the expense of a few evaluations of the error function. We illustrate the efficiency of this method by comparing it to the classical Riemann-Siegel formula.