The Riemann Zeta function $\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$ converges for $\operatorname{Re}(s)>1$. I am interested in some particular "irrational " values of it such as:

  • $\zeta(\pi)=1.176241738\ldots$,
  • $\zeta(e)=1.2690096043\ldots$,
  • $\zeta(\sqrt2)=3.020737679\ldots$,
  • $\ldots$

Are there closed form representations for these and constants? Are there formulas which consists of these constants?

  • 2
    $\begingroup$ The Riemann Zeta Function exists for all complex numbers except for $\;z=1\;$ ....Perhaps you meant the summatory form $$\sum_{n=1}^\infty\frac1{n^s}\;,\;\;\text{which converges for Re}\,(s)>1\; ?$$ $\endgroup$ – DonAntonio Sep 28 '13 at 16:32
  • $\begingroup$ Yes that. Specially !and @DonAntonio does the Riemann zeta function exists for $0$ too? $\endgroup$ – Shivam Patel Sep 28 '13 at 16:36
  • $\begingroup$ Yes Shivam: for any complex value except $\;1\;$ . $\endgroup$ – DonAntonio Sep 28 '13 at 16:42
  • 2
    $\begingroup$ In some sense, $\zeta(\sqrt{2})$ is already a closed form. $\endgroup$ – Hurkyl Sep 28 '13 at 17:32

There is no reason to suspect that these have a "closed form". There isn't even a known closed form for $\zeta(3)$...


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