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.)
As I said, the Riemann zeta function has its definition because it makes sense and the other doesn’t. Once you have a solid definition of ζ(-1), you could declare that 1+2+3+...=ζ(-1) and then you might be tempted to reverse the sign. But the zeta function was around for a century before Riemann encouraged people to emphasize the values that don’t make immediate sense.
You can do an awful lot just having it defined for real s>1. Euler used it to prove the infinitude of primes: ζ(1) is the harmonic series, thus infinite (or more precisely, an infinite limit as s approaches 1), but prime factorization expresses it as an product over primes, so there must be infinitely many primes to make it blow up. Moreover, this gives a better estimate of the density of the primes than Euclid’s proof. Then Dirichlet used it and related functions to prove that there are infinitely many primes satisfying reasonable congruences. (Exercise: use Euler’s technique to prove that there are infinitely many primes congruent to 1 mod 4 and infinitely many congruent to 3 mod 4.)
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?
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.
As I said, the Riemann zeta function has its definition because it makes sense and the other doesn’t. Once you have a solid definition of ζ(-1), you could declare that 1+2+3+...=ζ(-1) and then you might be tempted to reverse the sign. But the zeta function was around for a century before Riemann encouraged people to emphasize the values that don’t make immediate sense.
You can do an awful lot just having it defined for real s>1. Euler used it to prove the infinitude of primes: ζ(1) is the harmonic series, thus infinite (or more precisely, an infinite limit as s approaches 1), but prime factorization expresses it as an product over primes, so there must be infinitely many primes to make it blow up. Moreover, this gives a better estimate of the density of the primes than Euclid’s proof. Then Dirichlet used it and related functions to prove that there are infinitely many primes satisfying reasonable congruences. (Exercise: use Euler’s technique to prove that there are infinitely many primes congruent to 1 mod 4 and infinitely many congruent to 3 mod 4.)