# How to compute the alternating Hurwitz zeta function? Hi all!

What is the best way to compute the alternating Hurwitz zeta function with Sage?

Sage has an implementation of the Hurwitz zeta function, hurwitz_zeta(s,x), where s and x are complex, but not for the alternating Hurwitz zeta function.

There is the formula https://dlmf.nist.gov/25.11#E35 albeit with significant restrictions on the domain of s and x.

Which is a reliable way to implement hurwitz_alt_zeta(s,x) for general complex s and x, based on Sage functions?

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A project by Fredrik Johansson FunGrim: The Mathematical Functions Grimoire is to eventually transform automatically such formula into code which respects the restrictions on the domain, etc. I don't know if it is ready yet for what you want to do or if alternating Hurwitz zeta function is in there. See this recent preprint arXiv:2003.06181

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The right-hand side in https://dlmf.nist.gov/25.11#E35 looks like a perfectly fine global definition of the alternating Hurwitz zeta function, with the exception of special points. The restrictions Re(a) > 0 and Re(s) > 0 are just there to make the sum and integral well-defined.

The special points are s = 1 (where you need to compute the limit 0.5 * (digamma((a+1)/2) - digamma(a/2))) and the points where either Hurwitz zeta function is not defined with respect to a, namely nonpositive integer a (except when s is a nonpositive integer too).

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