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The Hurwitz zeta function:


reduces to $\zeta(s)$ when $a=1$ and to $(2^s-1)\zeta(s)$ when $a=\frac12$.

However, I stumbled upon a peculiar third connection:

$$\zeta_{H}(s,a) + \zeta_{H}(s,1-a)$$

that seems to exactly produce the non-trivial zeros of $\zeta(s)$,

when $a=\frac12$ (obviously), but also (and apparently only) when $a=\frac13, \frac14$ or $\frac16.$

Why does it only work for these values? Is there any reference to this in the literature?


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1 Answer 1

Found the answer, the question can be closed. answer

It boils down to:

$$\zeta_{H}(s,a) + \zeta_{H}(s,1-a) = \frac{4}{(2\pi)^{1-s}}\Gamma(1-s)C(1-s,a)$$

$$C(s,a)=\sum_{n=1}^\infty \frac{\cos(2n\pi a)}{n^s}$$

And $C(s,a)$ reducing to:

$a=\frac12 \rightarrow$ $(2^{1-s}-1)\zeta(s)$

$a=\frac13 \rightarrow$$\dfrac12(3^{1-s}-1)\zeta(s)$

$a=\frac14 \rightarrow$$2^{-s}(2^{1-s}-1)\zeta(s)$

$a=\frac16 \rightarrow$$\dfrac12(1-2^{1-s})(1-3^{1-s})\zeta(s)$

hence the non-trivial zeros.

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NOTE: there is an error in the formula above. It should be $\Gamma(1-s)\,\cos(\frac{\pi\,(1-s)}{2})$ (i.e. add the cosine). I don't want to bump this old post to the MO frontpage, so just added a warning here in the comment. –  Agno Sep 3 at 12:48

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