# 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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### Witten zeta function v.s. Riemann zeta function

From a talk, we learned that
The symplectic volume of the space $M$ of gauge equivalence classes of flat G connections is given by the “Witten zeta function”:
where we sum over irreducible ...

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### Irrationality of the values of the prime zeta function

Preamble: I asked this question on Math.SE with a bounty of 100, but received no replies. The bounty has now ended, and it was suggested I post this here instead.
Since Apéry we know that $\zeta(3)$, ...

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### Hodge theory (after Deligne)

In an interview with Deligne on the Simons Foundation website, I heard Robert MacPherson say that at the time Deligne's papers on Hodge theory were being published, the results seemed absolutely ...

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### Residues of Zeta-like Function

I'm looking for the residues of the following function $$s \mapsto\sum^\infty_{m,n =1} (m+n) \left[ amn + (m-n)^2 \right]^{-s}$$
at $s=\frac{1}{2}$ and $s=\frac{3}{2}$, where $a$ is some real positive ...

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### Analogues of the Riemann zeta function that are more computationally tractable?

Many years ago, I was surprised to learn that Andrew Odlyzko does not consider the existing computational evidence for the Riemann hypothesis to be overwhelming. As I understand it, one reason is as ...

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### Analytic Continuation of Zeta-like function

Reading a paper about eta invariants I came across a zeta-like function.
I'm looking for the analytic continuation of $$\sum_{k=1}^\infty k(k+a)^{-s}$$ at $s=0$, where $a$ is positive.
In the paper ...

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### What is the spectral interpretation of the arithmetic zeta function?

I recently stumbled upon the slides of a talk given by Kedlaya, in which the following appears:
For $X$ of finite type over $F_q$, a Weil cohomology theory, mapping $X$ to
certain vector spaces $...

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### Relation between infinite product and regularized product

For a positive sequence $0\le\lambda_{1}\le\lambda_{2}\le\cdots$, consider an infinite product
\begin{equation*}
\prod_{i=1}^{\infty}\lambda_{i}:=\lim_{n\rightarrow\infty}\prod_{i=1}^{n}\lambda_{i}...

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### Uniformity of the set of poles of Igusa local zeta functions

Let $Ω_p$ denote the set of the real parts of the poles of the Igusa zeta function of a polynomial $f∈\mathbb{Z}[X_1,…,X_m]$ (assume $f(0)=0$ so that $\Omega_p\ne \emptyset$) at the prime p. From ...

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### Comparisons of log canonical thresholds

Premise
Let $K$ be a field of characteristic zero and $f\in K[X_1,\dots,X_m]$. By Hironaka's theorem, there exists a log resolution (over $K$) of the ideal $(f)$. Let $\{(N_i,\nu_i)\}_i$ be the ...

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### Have partition functions of abstract simplicial complexes been examined?

Many complicated probability distributions arising in electrical engineering and machine learning have a simple expression as a sum of products that can also be encoded in a factor graph. The ...

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### Discrete approximation of Minkshisundaram-Pleijel zeta function?

I'm looking for some references on the following situation:
$S$ is a Riemannian surface, and $G_n$ is a sequence of metric subgraphs embedded on $S$. Let $\zeta_n$ be the zeta function of the ...

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### Kummer congruences for totally real number fields

There is a generalization of the Kummer congruences to totally real number fields with characters due to Deligne-Ribet. For example, see the exposition here, more precisely see Theorem 2.1.
What is ...

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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/...

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### Dirichlet series associated with polynomials

Yesterday, I was looking for references about meromorphic extensions of certain Dirichlet associated with polynomials. I was surprised to discover that much less than I previously thought was known.
...

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### 2-adic valuation of $L(0,\chi)$ for a Dirichlet character

Let $\chi : (\mathbb Z/f\mathbb Z)^\times \to K = \mathbb Q(\mu_{\phi(f)})$ be a primitive Dirichlet character. Assume moreover that it is not quadratic, that is, $\chi^2$ is not the trivial character....

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### Asymptotic for number of prime elements in a number field

Let $K$ be a number field and let $O_K$ its ring of integers. Identify $K$ with its image in $\mathbb{C}^{\text{Hom}_{\mathbb{Q}-\text{alg}}(K,\mathbb{C})}$, which we consider equipped with the $|| \...

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### Evaluating $\sum_{k=1}^{\infty} \frac{k}{2^k} \frac1{e^{t/{2^k}} + 1} $

If I arrive to calculate the sum $\displaystyle \sum_{k=1}^{\infty} \frac k{2^k} \frac{1}{\large e^{\frac t{2^k}} + 1} $ I think I can give a close form to the values $\zeta(1+\frac {2ip\pi}{\ln2})$ ...

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### Compensation by the residue of the zeta function

(Repost of a question from MSE, where it found no success)
Let $F$ be a global number field. Introduce a local quantity at every place
$$x_p = \frac{\zeta_p(1)}{\zeta_p(2)}$$
for instance. The ...

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### What are positive divisors of degree 2 on Elliptic Curve $y^2=x^3-x-1$ over $\mathbb{F}_3$?

Let (E) be an elliptic curve $y^2=x^3-x-1$ over $\mathbb{F}_3$, $a_0=1$, $a_n$ is the number of positive divisor of degree $n\geq 1$. $a_1$ in this case is the number of points of E, i.e., $a_1=1$ ...

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### Voronin universality of random analytic functions with nontrivial zeros on a line

Recently a certain random analytic function was defined by probabilists: in an appropriate sense the limit of characteristic polynomials of random unitary matrices. Associated functions for other ...

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### How does $\zeta^{\mathfrak{m}}(2)$ and relate to $\zeta(2)$?

EDIT There appears to be a numerical zeta function $\zeta(2)$ as well as at least two different "motivic" zeta function realizations (Betti and de Rham) $\zeta^{\mathfrak{m}}(2)$. The "period map" of ...

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### Values of Artin L-functions at negative integers

Let $F$ be a number field and $\chi$ a one dimensional Artin character. That is, it is a map $\chi: Gal(\overline F/F) \to \mathbb C^\times$ and let $L(s,\chi)$ be it's L-series.
What is known about ...

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### Fake integers for which the Riemann hypothesis fails?

This question is partly inspired by David Stork's recent question about the enigmatic complexity of number theory. Are there algebraic systems which are similar enough to the integers that one can ...

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### Zeta function of Abelian variety over finite field

Let $A/\mathbf{F}_q$ be an Abelian variety of dimension $g$. Suppose one knows $|A(\mathbf{F}_{q^n})|$ for all $1 \leq n \leq g$. Does one know then $\zeta(A,s)$ (equivalently, $|A(\mathbf{F}_{q^n})|$ ...

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### Properties of $\zeta(s)\zeta(2s)\zeta(3s)…$

Let's consider the Dirichlet series $f(s)=\sum_{n=1}^\infty a_n n^{-s}$, where $a_n$ is the number of non-isomorphic abelian groups of order $n$. Now $a_n$ is weakly multiplicative and $a_{p^k}=P(k)=$ ...

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### Critical values of L-functions and weights of Eisenstein Series

I have been reading Serre's paper on p-adic modular forms and there seems to be a connection between critical values of L-functions and weights of Eisenstein series in the following sense:
For the ...

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### Interchanging limit and infinite product in Euler product for Dedekind function s=1

For an quartic (non-Galois) CM-field $K$ I have factors $v_p$ and for every prime $p$ found the following relation
$$v_p={\frac {\prod_{\mathfrak{p}|p;\mathfrak{p}\subset\mathcal0_{K}}(1-N_{{K/{\...

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### Isospectrality, Gassmann-Sunada triples, and tensor products

It is well known that Gassmann-Sunada group triples can be used to construct isospectral manifolds, arithmetically equivalent number fields, etc. (Recall that a Gassmann-Sunada triple $(U,V,W)$ ...

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### Analytic extension of the Hurwitz ζ function

For the purpose of formalisation in a theorem prover, I am looking for a simple definition of the analytic extension of the Hurwitz ζ function $\zeta(s,q)$ valid for all $s\in\mathbb{C}\setminus\{1\}$ ...

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### How does the topology of the graphs' Riemann surface relate to its knot representation?

Let me give a worked-out example: The following cubic planar non-simple graph
$\hskip2.3in$
has the adjacency matrix $A=\pmatrix{0&3\\3&0}$. The graph has three faces, so the rank of $G$ is $...

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### $\zeta(0)$ of lichnerowicz operator on sphere quotients

Consider the lichnerowicz operator acting on symmetric traceless tensors on an even sphere $S^d$ with conical defect parametrized by angular deficit $\alpha$. Is there a simple way to understand/...

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### Characters of a quadratic extension and convergence

Let $F$ be a non-archimedean local field, $\chi$ a quasi-character of $F^\star$ and $\psi$ a positive character of $E^\star$. I would like to understand why the usual Rankin-Selberg zeta integrals ...

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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 ...

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### Hasse-Weil Zeta Functions & Fermats Last Theorem [closed]

Maybe this is a bit naive, but consider an equation of the form in Fermats Last Theorem $f_n(x,y,z) = x^n +y^n - z^n$.
Would it be possible to reprove Fermats Last Theorem by considering the Hasse-...

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### Estimation of the $k$-th derivative zeta function

When I was doing some task in number theory which involves bounds for the Riemann zeta function, I was stuck on the following question:
Let $\zeta$ be the Riemann zeta function and let $\zeta^{(k)}$ ...

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### Bernstein's theorem

In the book "An introduction to the theory of local zeta functions" prof. Igusa presents Bernstein's theorem as follows:
Let $K_0$ be a field, and write $K=K_0(s)$. Let $f\in K_0[x_1,\dots,x_n]\...

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### Does there exist a known Dirichlet series verifying all these conditions and have non trivial zeros off the critical line

Let $s=α+iβ$ be a complex number. Consider the Dirichlet series of the form $$f(s)=∑_{n=1}^{∞}(a_{n})/n^{s}$$
where $(a_{n})_{n≥1}$ is a real sequence.
We consider the class of Dirichlet series ...

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### Dirichlet series of a lattice $\sum_{a \in \Lambda^*} |\det(a)|^{-s}$

For a lattice $\Lambda$ of rank $n$ in $\mathbb{Q}^{n\times n}$ whose non-zero elements are inversible matrices, let $$Z(s,\Lambda) = \sum_{a \in \Lambda^*} |\det(a)|^{-s}$$
I wonder if (and how to ...

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### functional equation of $\zeta_X(s)$ in various settings

Say we have a set $X$ :
with a norm (possibly twisted by a character) and an unique factorization
$$\zeta_X(s) = \sum_{x \in X} |x|^{-s} = \prod_{P \in X} \frac{1}{1-|P|^{-s}}$$
from some additive ...

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### Regularisation of $\sum_{n=0}^\infty \frac{1}{(a+n^2)^p}$

As from the title, I am currently dealing with this sum
$\sum_{n=0}^\infty \frac{1}{(a+n^2)^p}$
in particular with $p=1/2,3/2,...$ (but once solved for $p=1/2$ one can derive wrt $a$ and find the ...

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### Behaviour of densities of places of finitely generated fields under specialisation

This question is a follow-up on question 2, posed in:
On the distribution of roots modulo primes of an integral polynomial
In appendix B of [1] by Pink, and in [2,3] by Serre, there are definitions ...

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### Selberg Zeta Function and Fenchel-Nielsen Coordinates

According to Uniformization theorem every compact Riemann surface $\Sigma$ of genus $g\ge2$ is isomorphic to a space that can be obtained by the action of a Fuchsian group on upper half plane $\mathbb{...

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### Relationships between different classes of L-Functions

There are many types of zeta (L) functions floating around. Lets consider
$\zeta_K(s)$ - the Dedekind Zeta Function of a number field
$L(\rho,s)$ - The Artin L-function $\rho:G_{\mathbb{Q}}\to GL_n(\...

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### Why are the formulations of Deligne-Ribet/Coates congruences for L-functions equivalent?

In Coates' $p$-adic L-functions and Iwasawa's theory, the first of his congruence hypotheses is that $\delta_n(\mathfrak{b},\mathfrak{c},\mathfrak{f})\in \mathbb{Z}_p$, where $\mathfrak{b},\mathfrak{c}...

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### The zeta function of a finite category determines the finite category?

In https://arxiv.org/pdf/1203.6133v3.pdf Kazunori Noguchi defines the zeta function for a finite category. It function is analogal to the Ihara zeta function of a graph. In http://emis.u-strasbg.fr/...

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### How to compute the following integral $I_{\alpha,\beta}$

We have the following identity (see Bateman, H. (1953). Higher Transcendental Functions [Volumes I], p. 25.)
$$(*)\quad \Gamma(\mu)\, \zeta(\mu,\nu) = \int_{0}^{1} x^{\nu-1} \,(1-x)^{-1} \Bigr(\log ...

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### Elementary symmetric functions of reciprocals of monic polynomials in function fields

Let $q$ be a prime power and $\mathbb{F}_q$ the field of cardinality $q$. Let $A = \mathbb{F}_q[T]$ and let $A_+ \subset A$ be the monic polynomials. Choose any ordering $<$ of $A_+$ and let $k$ be ...

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### Two questions about arithmetically equivalent number fields

Two algebraic number fields are said to be arithmetically equivalent if they
share the same Dedekind zeta function. If this is the case, they must have
certain invariants in common among which is the ...

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### Geometric and arithmetic Frobenius

I read in Serre's "Lectures on $N_X(p)$" that when $X$ is a scheme defined over $\mathbb{F}_q$ (a finite field), the geometric Frobenius $F: X \mapsto X$ is defined by fixing every element of the ...