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March 13

Product-based Generalizations of the Riemann ζ Function

The series expression of the Riemann zeta function gave rise, in light of the Mercator series for the natural logarithm, to the polylogarithm. However, no such analogous generalization is known for the Euler product expression of the same; in other words, why has no-one ever studied the expression $\prod _{p}\left(1-{\frac {z^{\color {blue}p}}{p^{s}}}\right)$ ? Or did they ? If so, what conclusions have they reached ? Likewise, did anyone ever study the infinite product expression $\prod _{p}\left(1-{\frac {z^{\color {blue}s}}{p^{s}}}\right),$ based on the similarity to the famous Euler product for the sine or sinc function (see Basel problem) ? And if not, is there anything noteworthy that can be said about the two ? — 79.113.197.194 (talk) 02:38, 13 March 2017 (UTC)[reply]

Not my area, but rather than let the question go unanswered... My understanding is that the reasons the Riemann zeta function is useful are first, it can be written in terms of the primes, and second, the functional equation relates it to well understood functions. So you need to find the analog of the functional equation for the functions you're defining. As to what has been investigated, it's only possible to say what has been published; you can't tell what ideas have been tried but have gone nowhere and so weren't published. (It seems like it would save a lot of duplicated effort to tell people what did't work but there you go.) There are quite a few generalizations of the Riemann zeta function which have been published, so the functions you defined might be among them (again, not my area). You might start with the section on generalization in the article. --RDBury (talk) 11:27, 15 March 2017 (UTC)[reply]