manuscripta mathematica

, Volume 101, Issue 3, pp 385–399

Approximation of singular series and automata


  • Pieter Moree
    • Mathematical Institute, University of Leiden, P.O.Box 9512, 2300 RA Leiden, The Netherlands. e-mail:

DOI: 10.1007/s002290050222

Cite this article as:
Moree, P. manuscripta math. (2000) 101: 385. doi:10.1007/s002290050222


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.

Mathematics Subject Classification (1991):11Y16, 11Y60, 68Q45

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© Springer-Verlag Berlin Heidelberg 2000