/sci/ - Science & Math

[Anonymous No.9752115]

The function that is called local zera function in the Weil conjectures has properties (the zeros theorem on the critical line) akin to the Riemann zeta function. But it's not like the Riemann zeta function has a nice representation
[math] \zeta(s) = \exp(...) [/math]
Right?

I'm asking because I'm looking e.g. at the Lefschetz zeta function, which looks like that of the Weil conjecture, but any connection to Riemanns seems lost

https://en.wikipedia.org/wiki/Lefschetz_zeta_function
https://en.wikipedia.org/wiki/List_of_zeta_functions
https://en.wikipedia.org/wiki/Weil_conjectures#Statement_of_the_Weil_conjectures

[Anonymous No.9752142]

There's this:
https://math.stackexchange.com/q/1791777
[math] \zeta(s) = \exp\left(\sum_{n=2}\frac{\Lambda(n)}{\log(n)} n^{-s} \right)[/math]

and along the lines of the Weil conjectures, you might be interested in this speculation:
https://mathoverflow.net/a/23423/123345
>Now if X=SpecZ were somehow a smooth projective variety of dimension 1 over SpecF1, then ζX(s) would be the Riemann zeta function, and...

[Anonymous No.9752169]

>>9752142
Thanks for the reply. Yeah I had played around a bit after asking and, using the Euler product and some basic manipulations, ended up with

[math] \zeta (s) = \exp \left( \sum_p \sum_k \dfrac { p^{ks} } {k} \right) [/math]

My question is motivated by browsing through pic related. They have some (transition) matrix [math] A [/math] that upon application makes points move through space and then they study fixed points of jumps [math] A^k [/math]. I suppose this is how you end up looking at (powers of) eigenvalues [math] a_i^k [/math] which in the number theory case are something like [math] p_i^k [/math], where p_i are all the primes. Then if you define a "zeta function" as the exponential over a sum of some fixed point counting object, you end up with this function being a product over your solutions and that's akin to the Euler product. But what's also nice is that in that case, the product is a determinant over some operator related to A.

[Anonymous No.9752177]

Oh, reading that comment below in pic related I suppose that the fact that the Riemann (and local zeta-function) have "two exponents" k and s
[math] p^{ks} [/math]
can be read as one of them being the exponent in the Frobenius map. I.e. if you want to shoehorn the rading of the zeta function being a series over powers of a map, then the endomorphism [math] q \mapsto q^n [math] in F_n is what you should do.