Questions tagged [zeta-functions]
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.
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Statistical meaning of zeros of zeta functions
To simplify the discussion, I consider the Riemann zeta function but my question should make sense for most zeta functions (Selberg, algebraic varieties over finite fields, ...).
Let $\zeta(s) := \...
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Čech cohomology satisfies descent if the underlying spectrum universally bounded from below
Let $X\longrightarrow\operatorname{Spec}(k)$ be an étale scheme of finite relative dimension $d$, $U\subseteq X$ an affine open subscheme, $\mathcal{U}$ a hypercovering consisting of affine schemes.
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Isomorphisms of sheaves of Abelian groups in Hesselholt's "Topological Hochschild Homology and the Hasse-Weil Zeta function"
This is from §5, p. 13, of Hesselholt's "Topological Hochschild Homology and the Hasse-Weil Zeta function".
The author claims there is a family of isomorphisms
\begin{equation*}
\phantom{\...
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Is there an algebraic notion of two primes being close?
I apologise in advance for the vagueness of the question below. I am not at all an expert in algebraic geometry, it might be that the question will come across as very naive, sorry!
I was wondering ...
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Distribution of lengths of closed geodesics
(I am trying to get a sense of what the state of the art is regarding the distribution of the length spectrum of a closed surface of negative curvature, I am curious about any good reference/open ...
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Dynamical systems of twisted Ihara zeta functions on graphs
A twisted Ihara zeta function of a finite connected graph $\Gamma$ is sensitive to a quantity known as holonomy. The zeta function can be defined as:
$$\zeta_{\Gamma}(u, \rho) := \prod_{[P]} \left(1 - ...
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Dirichlet series from second differences of recursively summed prime gaps
I'm investigating a Dirichlet series built from a recursively summed and differenced sequence of prime gaps.
$\text{Let } g_n = p_{n+1}-p_n$, denote the prime gaps. From these, construct:
$$S_0(n)=g_n,...
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Shared large partial quotient and modular-like relations in continued fractions of ζ(3)/(m² log π)
Update: Two additional unusual continued fraction behaviors have been observed for expressions involving zeta(3) and log(pi) and are documented in the second script below.
I've recently encountered ...
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Why are there no L-factors corresponding to the infinte places for $\zeta$ functions over $\mathbb{F}_{p^{n}}$?
Deninger has constructed $L$-factors $\zeta_\infty(s) := 2^{-1/2} \pi^{-s/2} \Gamma(s/2)$, that are to be interpreted as "$L$-factors at infinity".
The conjectural "Deninger Trace ...
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Graphs and reflection-induced color symmetries
Let $\Gamma = (V, E)$ be a finite, planar labeled, undirected, tailless, topological multigraph ($4$-regular) arising from a $z=0$ slice of a stratified foliated manifold. The graph admits a ...
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Understanding choice of smoothing function in Titchmarsh
I am currently reading Chapter 4 of Titchmarsh's Theory of the Riemann zeta function, about Approximate Formulae. The following theorem is given.
Theorem: We have \begin{equation*}
\zeta(s)^k=...
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Is there a "multiple" Igusa zeta function?
Let $[K:\mathbb Q_p]<\infty, R=\mathcal{O}_K$, and $\pi$ a uniformizer. The Igusa zeta function is defined for a Schwartz--Bruhat function $\phi:K^n\to \mathbb C$, a character $\chi:R^\times\to \...
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Relation between left ideal and right ideal zeta functions of orders
We try to keep things as simple as possible. Let $A$ be a $\mathbf{Q}$-algebra which is a simple ring of dimension $n$ over $\mathbf{Q}$. Let $\mathfrak{O}\subseteq A$ be an order, i.e. a subring ...
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The Hurwitz zeta function on the line $\mathrm{Re}(s) = 1$
For $\mathrm{Re}(s) > 1$ and $0 < a \leqslant 1$, let $\zeta(s,a) = \sum_{n \geqslant 0} 1/(n+a)^s$ denote the Hurwitz zeta function. As a function of $s$, $\zeta(s,a)$ has a meromorphic ...
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Equivalence of an iterated integral and a multiple zeta value
As I understand, the multiple zeta function is related to Chen's iterated integral in the following way:
$$
\zeta(s_1, s_2, \dots, s_k) = \int_{0}^1 \frac{1}{t_1^{s_1 - 1}} \int_{0}^{t_1} \frac{1}{1 - ...
