Approximation of singular series and automata

Abstract

A constant of the form Πp h(p), where the product ranges over all sufficiently large primes p and h is rational, is an example of a singular series. We show that this type of singular series can be expanded in the form Π∞k=2 ζ(k)-ek, where ζ denotes the zetafunction and ek is an integer and use this to numerically approximate them. Gerhard Niklasch in an appendix describes how to obtain more than 1000 decimal accuracy. In some cases the coefficients ek turn out to be related to conjugacy classes of primitive words in cyclic languages.