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In particular for two real numbers $a$ and $b$, I'd like to know if there are formulas for $\zeta (a+b)$ and $\zeta (a-b)$ as a function of $\zeta (a)$ and $\zeta (b)$.

The closest I could find online is a paper by Harry Yosh "General Addition Formula for Meromorphic Functions Derived from Residue Theorem" in some little known journal, but unfortunately I have no access to it and don't know if it would answer my question. Maybe this is well-known and I didn't search correctly...

Any help appreciated, thanks!

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    As far as I know there are formulas relating, $\zeta(s)$ with $\zeta(s + 1), \zeta(s + 2), \zeta(s + 3), ...$. However they are not particularly deep nor useful. I think you can find them in the first chapters of Titchmarsh's book on the Riemann zeta-function.2012-11-12

3 Answers 3