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It is becoming increasingly clear that the expression $~\displaystyle\prod_{n\in\mathbf M}\left[1-\frac{z^s}{(n-a)^s}\right],~$ with $~|\mathbf M|=\aleph_0,~$

$a\not\in\mathbf M,~$ and $~\Re(s)>1,~$ lends itself as a natural generalization of all four classes of functions

mentioned in the title, since

  • for natural values of n and s, it yields a (finite) product of $\Gamma$ functions, which, in certain cases (depending on the parity of s, and on whether a is either an integer or a half-integer), can be further simplified to a (finite) product of trigonometric and/or hyperbolic functions.

  • for prime values of n, with $a=0,$ and $|z|=1,$ it yields a (finite) product of $\zeta$ functions.

My actual question would be what exactly happens in the former case for $\color{blue}{s\in\mathbb R\setminus\mathbb N},$ and in the latter for $\color{blue}{|z|\not\in\{0,1\}}.~$ But just in case answering this might prove too vast of an undertaking, I've decided it might be best to try and simplify things a bit by splitting it into two distinct sub-questions, asking specifically for the evaluation of $~\displaystyle\prod_{n\in\mathbb N^\star}\left(1+n^{-3/2}\right),~$ and $~\displaystyle\prod_{p\in\mathbb P}\left(1+\frac{2^2}{p^2}\right).~$

The very first thought crossing my mind is whether any of the two (or perhaps something very similar to them) has ever been mentioned in the literature. The other one would be to inquire for acceleration methods pertaining to the (rather thorny) issue of their numerical computation, since all mathematical software I can think of (Mathematica, Maple, PARI/GP) experience significant hurdles whilst trying to determine even the first few digits of their decimal representation. (Thus, all I know so far about the first product, for instance, is that it is approximately equal to 9.20). Any help would be deeply appreciated !

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    $\begingroup$ I assume this question keeps on acquiring downvotes for a reason. However, that reason eludes me. Is this question not research level ? If so, then this means that someone has already researched it; in which case, what were their conclusions ? Or is it merely because the users of this site prefer their questions from other fields of mathematics ? In that case, it would be helpful to stipulate such matters clearly in the site's description, since downvoting what appear to be perfectly legitimate questions seems off. $\endgroup$
    – Lucian
    Commented Jan 8, 2017 at 10:24
  • $\begingroup$ (except when $z = \pm 1$) $\log \prod_p (1-z p^{-s}) = -\sum_p\sum_{k \ge 1} \frac{z^k p^{-sk}}{k}$ and $\sum_p p^{-s} = \sum_{n=1}^\infty \frac{\mu(n)}{n} \log \zeta(ns)$ so it is an infinite linear combination of $\log \zeta(ns)$ $\endgroup$
    – reuns
    Commented Jan 8, 2017 at 22:26
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    $\begingroup$ Lucian, I didn't downvote, but I might suspect that a number of people (as would I) would not see as promising a viewpoint that succeeds in putting a too-large variety of things under one umbrella... the reason(ing) being that then whatever might be true for all cannot be too deep for individuals. Yes, such opinions are a matter of judgement. As a tangible example, any proof mechanism that purports to prove RH but also accidentally would apply to Epstein zetas cannot possibly be correct, because many of the latter are known to have many off-line zeros, even under GRH, etc. $\endgroup$
    – paul garrett
    Commented Feb 18, 2017 at 23:09
  • $\begingroup$ @paulgarrett: There is certainly much truth to that. However, the relation between trigonometric and hyperbolic functions, known since antiquity (conic sections), has been strengthened by Euler's formula, and today we possess a very powerful formal mechanism of elegantly manipulating the two in a unifying and coherent manner; the relation between the $\Gamma$ and (co)sine function is given by two reflection formulas, which are a direct implication of the latter's infinite product expression, also due to Euler; and the relation between the $\Gamma$ and $\zeta$ functions dates back to Riemann. $\endgroup$
    – Lucian
    Commented Feb 18, 2017 at 23:48
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    $\begingroup$ Lucian, indeed, the work of Iwasawa and Tate c. 1950 shows that Gamma is a "local archimedean" version of a "global" (=adelic) set-up that produces the completed zeta function. But/and given the 150 years' difficulty of understanding zeta, one might reasonably hesitate to think that much will be learned about it by a very broad approach. Mandelbrojt's, Boas', deBranges', and others' substantial ideas about entire functions didn't see to make progress on zeta, for example. But maybe that's not your principal goal... $\endgroup$
    – paul garrett
    Commented Feb 19, 2017 at 0:04

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