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I have been reading about the RH recently and I understood most of it until now. However, the biggest problem I'm having is to know what are the forms of the Riemann zeta function for the 3 main regions in the complex plane, $\Re(s) <-1$, $0 \le \Re(s) < 1$, and for $\Re(s)> 1$. Also, I have seen that zeta can be defined as the following integral.

$$ \frac{1}{\Gamma(s)}\int_0^\infty \frac{x^{s-1}}{e^x-1}\, \mathrm{d}x,$$

Is the zeta function defined on the entire complex plane, except $1$? And about the other ones? Also, are there other integrals for zeta, some whose limits of integration are different than zero and infinity?

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    $\begingroup$ Yes. The domain of $\zeta$ is $\mathbb{C} \setminus\{1\}$ $\endgroup$
    – Mason
    Commented Jan 6, 2020 at 21:31
  • $\begingroup$ Oh, thanks again, Mason $\endgroup$
    – Mr. N
    Commented Jan 6, 2020 at 21:33
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    $\begingroup$ Hard to know what you've already seen. The zeta function is well-studied and there are many different formulations of it. Wikipedia, Wolfram alpha, Wolfram Alpha Again $\endgroup$
    – Mason
    Commented Jan 6, 2020 at 22:09
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    $\begingroup$ I am not really sufficiently knowledgeable in this topic to recommend a text. And it depends greatly on your mathematical background+interests. Best I can do for you is defer. $\endgroup$
    – Mason
    Commented Jan 7, 2020 at 3:16
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    $\begingroup$ Different expression for the Zeta function exist for those regions. This is because of analytic continuation, since the Dirichlet series does not converge for any $\Re(s)<=1$. In those regions you find "extended" versions of $\zeta$. $\endgroup$
    – Klangen
    Commented Jan 7, 2020 at 7:21

1 Answer 1

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From the picture in the youtube video by numberphile starting at 11:44

$$\zeta(s)=\sum\limits_{n=1}^{\infty}\frac{1}{n^s}=\prod\limits_{p \text{ prime}}\frac{p^s}{p^s-1}, \;\;\;\;\; \Re(s)>1$$

$$\zeta(s)=(1-2^{1-s})\sum\limits_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^s}, \;\;\;\;\; 0<\Re(s)<1$$

$$\zeta(s)=\left((2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\right)\zeta(1-s),\;\;\; \Re(s)<0$$

There is this integral relation in the OEIS: $$\left(1-\frac{1}{2^{s-1}}\right) \zeta (s) \Gamma (s+1)=\int_0^{\infty } \frac{1}{e^{x^{1/s}}+1} \, dx, \;\;\;\; \Re(s)>0$$

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  • $\begingroup$ Oh thanks. I had already seen it, before asked. My question was related to the integral forms of Zeta and their domain. By the way, do you know any integral form for Zeta in the critical strip? $\endgroup$
    – Mr. N
    Commented Feb 14, 2020 at 17:07
  • $\begingroup$ You might find something in the OEIS if you look at the decimal expansion for Zeta[2] oeis.org/A013661 $\endgroup$
    – Mats Granvik
    Commented Feb 14, 2020 at 17:21
  • $\begingroup$ Also try these: functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/07 $\endgroup$
    – Mats Granvik
    Commented Feb 14, 2020 at 17:41
  • $\begingroup$ Thanks. The one in the OEIS looks pretty good! $\endgroup$
    – Mr. N
    Commented Feb 14, 2020 at 18:17
  • $\begingroup$ And then there is also the truncated Euler Maclaurin formula for the Riemann zeta function if you like simplicity: $$\zeta(s)=\lim_{k\to \infty } \, \left(\sum _{n=1}^k \frac{1}{n^s}+\frac{1}{(s-1) k^{s-1}}\right), \;\;\;\; \Re(s)>0$$ But that is of course not an integral. $\endgroup$
    – Mats Granvik
    Commented Feb 14, 2020 at 18:40

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