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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. –  blabler Nov 12 '12 at 1:54

3 Answers 3

Almost certainly, the answer is no. Consider $a$ with real part in the interval (1/2,1). The Riemann hypothesis states that $\zeta(a)\not=0$. If $\zeta(a)$ were a function of $\zeta(2+a)$ and $\zeta(2)$ then this would convert the RH into a statement about the values of $\zeta(s)$ on $5/2<\Re[s]<3$, which would be just too easy.

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There is a functional equation relating values of $\zeta$ at $s$ and $(1-s)$.

If there were formulas relating values along a continuous family of 1-dimensional curves, such as $x + y = C$, one would get a differential equation for $\zeta$, or some comparably strong constraint. It is known that $\zeta$ does not satisfy any ODE's with algebraic functions as coefficients. Of course there could be gamma functions or other more complicated coefficients but prospects for this kind of additional structure in $\zeta$ seem dim. There are no extra functional equations for finite field zetas, for example.

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I recall a formula of Ramanujan that expresses an L-function with divisor function coefficients in terms of zeta functions: $\sum_{n=1}^\infty \sigma_a(n)\sigma_b(n) n^{-s} = \zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)/\zeta(2s-a-b)$. –  chroma Dec 1 '10 at 18:53
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@chroma: The L-function on the left side depends on all three of (s,a,b), so it is more a "functional formula" than a functional equation. –  T.. Dec 1 '10 at 19:08
    
@chroma +1 well said. –  mick Jul 15 '13 at 20:43

I've never seen any such formulas. If there were any, the would most likely be listed in the Zeta section at functions.wolfram.com.

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