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Results tagged with zeta-functions
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user 1384
Zeta functions are typically analogues or generalizations of the Riemann zeta function. Examples include Dedekind zeta functions of number fields, and zeta functions of varieties over finite fields. They are typically initially defined as formal generating functions, but often admit analytic continuations.
18
votes
What is the difference between a zeta function and an L-function?
Although no-one else seems to have suggested this, my personal take on this is that there's no difference whatsoever. Sure, if people start talking about "Dedekind zeta functions" or "Artin $L$-functi …
- 41.4k
8
votes
Can the failure of the multiplicativity of Euler factors at bad primes be corrected?
James: given that no-one else answered this yet, let me just make some naive comments that you probably know already.
Of course the problem is that if $I_p$ isn't acting trivially, then "taking $I_p …
- 41.4k
11
votes
The class number formula, the BSD conjecture, and the Kronecker limit formula
Not a great answer, but some comments that hopefully push in the right direction.
For a number field $K$, there is naturally a finite-dimensional complex vector space associated to it, namely the spa …
- 41.4k
8
votes
Why are functional equations important?
I am surprised no-one seems to have mentioned one key use for functional equations: they are a key input in converse theorems. If you have a power series in q that you're trying to show is a modular f …
- 41.4k
34
votes
2
answers
3k
views
The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.
I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt correc …
- 41.4k
20
votes
3
answers
2k
views
Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number...
Let $M$ be the splitting field of
x^8 + 3*x^7 + 13*x^6 + 17*x^5 + 45*x^4 + 37*x^3 + 11*x^2 + 112*x + 108
over the rationals. If I've understood some tables correctly, the splitting field is (of cou …
- 41.4k