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The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line (and outside of critical strip)?

Conjectured irrationality, mysteriously neat expressions or something of the like.

Edit. Since the question is necessarily broad, I'll try to make it a bit more clear:

Is there any $s\in \mathbb{C}$, $\mathfrak{I}(s)\neq0$, $\mathfrak{R}(s)>1$ such that $s$ or its real/imaginary components are suspected to be transcendental, or to be give any new arithmetical information about $\mathbb{Q}$?

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  • $\begingroup$ At least, the motivation (pun intended) between the formulae expressing the values at integers seems to disappear, so if there exists interesting expressions of the values elsewhere, it will be for completely different reasons of which I am unaware. $\endgroup$
    – Olivier
    Commented Sep 26, 2014 at 12:55
  • $\begingroup$ Nice observation. There don't seem to be any, ignoring of course the recurrence relation between $\zeta(s)$ and $\zeta(1-s)$ or tricks like expressing $\displaystyle \zeta(s)= \left(\zeta_H(s,a)-\zeta_H(s,a+1)\right)\,\sum_{k=1}^{a} \zeta_H\left(s,\frac{k}{a}\right)$ in terms of the Hurwitz zeta function that is valid for all $a \in \mathbb{N}$ and also for all $s \in \mathbb{C}\,/1$. Both formulae do "happen arithmetically outside the real line" however, there doesn't seem to be any result that links back to attributes of 'human reality' like integers, rationals or irrationals like $\pi$. $\endgroup$
    – Agno
    Commented Sep 26, 2014 at 17:30
  • $\begingroup$ @Olivier. Is there a good explanation of why nothing motivic could (should?) happen outside the integers? That would be an interesting partial answer. $\endgroup$
    – Myshkin
    Commented Sep 27, 2014 at 8:10
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    $\begingroup$ @Agno. Given the rich information contained in the integers and the critical line, the rest seems like a suspicious waste of space. $\endgroup$
    – Myshkin
    Commented Sep 27, 2014 at 8:18
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    $\begingroup$ "seems like a suspicious waste of space"... That is sheer poetry, Myshkin :-) Your question did provoke a philosopical thought that I posted as an answer. $\endgroup$
    – Agno
    Commented Sep 27, 2014 at 16:16

2 Answers 2

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This answer is more philosophical than mathematical, but I decided to take a risk today...

One could also conjecture that actually all interesting information is only contained in the non-trivial zeros in the critical strip. The logic would be that on top of encoding the distribution of primes, the non-trivial zeros $\rho$ in the Hadamard product:

$$\displaystyle \zeta(s)=\frac{\pi^{\frac{s}{2}}\,\prod_{\rho}\left(1-\frac{s}{\rho}\right)}{2\,(s-1)\,\Gamma(1+\frac{s}{2})}$$

also encode (with some help) all the "trivial" results at the integers.

The distinction between "trivial" and "non-trivial" zeros, is a label driven by our human perception of 'what we can simply explain' and 'what we can't easily understand'. So, there could be many interesting "abstract" results outside the strip, that don't yet resonate in any way with our current state of (even the abstract) mathematical knowledge. Poetically labelling this domain as "a suspicious waste of space" would hopefully one day turn into "you'd ain't seen nothing yet".

The more I understand about analytic continuation, the more I also start to wonder whether a more appropriate term would actually be: 'meaningful analytic discontinuation'. My logic is that an analytically continued function like $\zeta$ is valid in a much wider and richer domain, than the limited domain of convergence (e.g. $\Re(s)>1$) that we typically start from and frame up in terms of infinite sums, integrals and/or (Euler) products based on powers of integers/primes.

Having worked quite a bit with the Zeta-function, I actually start to truly appreciate the beauty (or better "L'affreuse beauté" as the French would say) of the non-trivial zeros and them possible all lying on the critical line. Yes, I start to like them even more than the primes themselves (I know, that is a very subjective statement). This feeling is reinforced by the Hadamard product above, that also allows the Euler product of primes to be encoded by the $\rho's$ (no chance you could do the same with the primes, although of course they deserve credit for encoding all the integers :-) ).

It might be similar to the journey physics went through (I know, they invented Zeta regularization. A method that is not considered correct in pure mathematics), where many scientists still try to align the counter intuitive aspects of Quantum Mechanics with the 'classical limit' of the 3D world we live in.

Speaking of physics, we could also slightly rephrase your question as: "Whether any special values of $\zeta$ exist outside the real line and the critical strip, show up in physics?" Again the answer seems no, since $\zeta(-1)=-\frac{1}{12}$ (again an integer) shows up in the Casimir force (although an answer to this question suggests that $\zeta'(-\frac12)$ also plays a role in this effect) and potentially the distribution of the imaginary parts of the $\rho's$ (the strip again) being correlated to eigenvalues of random GUE matrices used in Quantum Mechanics. I don't know of any other examples.

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  • $\begingroup$ This is precisely the kind of answer I didn't know I was looking for. I'm really glad that you took the risk! I haven't quite digested it, but I have some follow-up questions alredy: You mention the non-trivial zeros encoding (via Hadamard product) "all trivial results at the integers". Do you mean anything else beside the negative even integers and 1? Since, well, we can't count with a complete list of zeros on the critical strip, the question would be, how much the arithmetic data at the integers (conjecturally) tells us about the distribution of the zeros as encoded by the Hadamard product? $\endgroup$
    – Myshkin
    Commented Sep 28, 2014 at 9:04
  • $\begingroup$ I don't understand the remark about "meaningful analytic discontinuation", I'd appreciate any further explanation. $\endgroup$
    – Myshkin
    Commented Sep 28, 2014 at 9:40
  • $\begingroup$ "Do you mean anything else beside the negative even integers and 1?". Yes, since the $\Gamma$ induces the zeros at the negative even integers and $(s-1)$ gives the pole at 1. We indeed cannot count with all zeros in the strip, however it appears to be all we have got. Take for instance $\prod_{\rho}\left(1-\frac{2}{\rho}\right) = \frac{\pi}{3}$. I therefore could approach $\pi$ by multiplying more and more $\rho's$, but I am not aware of any method to do it vice versa and tell anything about their distribution or prove that they all (or even one!) lie on the critical line. $\endgroup$
    – Agno
    Commented Sep 28, 2014 at 10:33
  • $\begingroup$ @Myshkin. In the examples of analytic continuation that I know of, the starting point always are series composed of integers/rationals (i.e. "meaningful", natural things). Series are then found to converge in a limited domain and when divergence is encountered, attempts are made find their unique analytic continuation. It always seems to happen this way. A MO-question could be: Is there an example in complex analysis, where we started with a holo/meromorphic function valid across $\mathbb{C}$, and then only later discovered it could be 'meaningfully discontinued' in a smaller domain? $\endgroup$
    – Agno
    Commented Sep 28, 2014 at 10:58
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There is a well known conjecture that the ordinates of the non-trivial zeros of $\zeta$ are $\mathbb{Q}$-linearly independent.

There are two major motivations for this conjecture. First all numbers which are not obviously rational are irrational, second linear independence together with RH implies that the suitably renormalized error term in the prime number theorem behaves like a sum of independent random variables, so we can easily compute its distribution.

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    $\begingroup$ I'd say ordinates rather than abscissae. $\endgroup$
    – Sylvain JULIEN
    Commented Sep 27, 2014 at 10:10
  • $\begingroup$ I was looking outside of both the real line and the critical strip. Hopefully now the title is more clear. $\endgroup$
    – Myshkin
    Commented Sep 27, 2014 at 10:41
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    $\begingroup$ +1: you had me at "all numbers which are not obviously rational are irrational". $\endgroup$
    – user41593
    Commented Sep 27, 2014 at 16:25
  • $\begingroup$ @Sylvain Julien: Thanks, corrected that. $\endgroup$
    – Jan-Christoph Schlage-Puchta
    Commented Oct 13, 2014 at 8:16

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