All Questions
7 questions
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Relating the multiplicative Fourier transform and the derived characteristic polynomial
(Tuesday, Sept 5:) For a number field $Fˣ$ and a number ring $Oˣ$ it is common to define:
$Z(f,χ) = ʃ_{Fˣ} f(x) χ(x) dˣ x$
$g(ω,ψ) = ʃ_{Oˣ} ω(x) ψ(x) dˣ x$
where $dˣx$ is the multiplicative Haar ...
user30211
2
votes
0
answers
104
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The Guinand-Weil explicit formula for Hecke characters
The Guinand-Weil formula for the Riemann zeta function is
\begin{aligned}&\Phi (1)+\Phi (0)-\sum _{\rho }\Phi (\rho )\\={}&\sum _{p,m}{\frac {\log(p)}{p^{m/2}}}{\Big (}F(\log(p^{m}))+F(-\log(p^...
- 9,501
15
votes
1
answer
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How do functional equations for zeta functions arise from the structure of a homology group?
I have read in various disparate sources that certain zeta functions satisfy functional equations as a consequence of some structure on some homology group. Here is an example of a quote in this ...
- 1,231
14
votes
2
answers
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What are zeta functions good for?
I know a couple of answers to the above question:
They can be used for point counting over finite fields/estimating the distribution of primes in characteristic 0.
There are various conjectures/...
9
votes
1
answer
1k
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Functional equation Dedekind zeta function
I'd like to know to what point is it possible to generalize
this method
for obtaining the functional equation for the Dedekind zeta function $\zeta_K(s)$ of a number field ?
Let $\mathfrak{C}$ be ...
- 3,403
3
votes
1
answer
367
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Weil Conjectures Analog for Multivariate Zeta Functions
We know that the Riemann zeta function can be generalized to multivariate zeta functions.
Is there a multivariate analog of the Weil conjectures?
- 13.9k
20
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3
answers
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Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field.
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 ...
- 41.4k