What is your complaint about Zeta? That it is the sum of n^-s, rather than the sum of n^s? It's the one that converges. Or are you bothered that zeta(-3) is rational, while zeta(3) is irrational?
A function that fills in the gaps between factorials seems useful.
Maybe useful for some purposes. Maybe that would be a good function to have when defining the Beta distribution, though there are other reasons for the normalization there.
But in the context of the Riemann Zeta function (which is the context you have suggested), that is not at all the purpose of the Gamma function. Its role is as the Mellin transform of the exponential function. The Zeta function itself is a Mellin tranform and two interact well because of their common origin. Of course, that pushes back the question to why the Mellin tranform has a -1. What it really has is a dx/x. This measure is invariant under scaling, just as dx is invariant under translation. Indeed, the measures correspond under the exponential change of variables.
(In fact, that is closely related to a justification for the normalization of the Beta distribution. B(0,0) is the measure invariant under logistic transformations; B(p,q) is the posterior after seeing p,q observations.)
Great answer, thanks.
Yes, my shallow, uninformed by higher maths complaint about the zeta function is that it sums n^-s instead of the simpler n^s.
This thread is for asking any questions that might seem obvious, tangential, silly or what-have-you. Don't be shy, everyone has holes in their knowledge, though the fewer and the smaller we can make them, the better.
Please be respectful of other people's admitting ignorance and don't mock them for it, as they're doing a noble thing.
To any future monthly posters of SQ threads, please remember to add the "stupid_questions" tag.