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 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 $e_k$ turn out to be related to conjugacy classes of primitive words in cyclic languages.
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