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How to compute the alternating Hurwitz zeta function?

asked 2020-05-06 15:23:44 +0200

Peter Luschny gravatar image

updated 2021-07-15 08:24:21 +0200

FrédéricC gravatar image

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

Sébastien gravatar imageSébastien ( 2020-05-06 18:14:41 +0200 )edit

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answered 2020-05-07 15:26:51 +0200

fredrik gravatar image

updated 2020-05-07 15:27:37 +0200

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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Asked: 2020-05-06 15:23:44 +0200

Seen: 101 times

Last updated: May 07 '20