Approximation of singular series and automata

Abstract:

A constant of the form , 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 , where ζ denotes the zeta-function and e _k 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 $e_k$ turn out to be related to conjugacy classes of primitive words in cyclic languages.