Computational methods and experiments in analytic number theory

Abstract

We cover some useful techniques in computational aspects of analytic number theory, with specific emphasis on ideas relevant to the evaluation of L-functions. These techniques overlap considerably with basic methods from analytic number theory. On the elementary side, summation by parts, Euler Maclaurin summation, and Mobius inversion play a prominent role. In the slightly less elementary sphere, we find tools from analysis, such as Poisson summation, generating function methods, Cauchy's residue theorem, asymptotic methods, and the fast Fourier transform. We then describe conjectures and experiments that connect number theory and random matrix theory.