# How is the analytical continuation for the HurwitzZeta function implemented?

Following up on this question, I am trying to understand the implementation details of the HurwitzZeta[x,y] function in Mathematica, particularly when the first parameter is less than 1. According to the classical definition of the Hurwitz Zeta function, it should diverge for arguments where the first parameter is less than 1. Yet, when I use Mathematica to evaluate expressions like HurwitzZeta[1/3, 1] // N, I get a finite, well-defined value (in this case, "-0.97336").

I have consulted Wolfram's MathWorld page on the Hurwitz Zeta function,(https://mathworld.wolfram.com/HurwitzZetaFunction.html, and it's clear that Mathematica uses an analytic continuation to Hurwitz Zeta to compute these values. However, the specific method used for this continuation is not explained.

Can anyone provide insight into the specific formulae that Mathematica uses to compute these values?

• HurwitzZeta is defined with the help of Zeta and an example of series of Zeta[z] for 0 < Re[z] < 1 representing the unique analytic continuation from Re[z] > 1 I included in my answer. There are infinitely many series reperesentations of Zeta, they depend on the initial point and the radius of convergence. Wolfram documentation doesn't inform about specific choice of implementation, you can't get more information besides e.g. tutorial on Mathematical Functions. Jul 12, 2023 at 16:50
• For the rest of the complex plane (Re[z] < 0) the system uses also the functional equation $\zeta(s) = 2^s \pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s)$. Jul 12, 2023 at 17:00