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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Are there any zeta functions with concurrent derivative shifts in multiple variables?

Expressions for rational zeta series have been obtained by considering the Taylor series of zeta functions. For instance, one has \begin{align}\zeta(s,x+y) &= \sum_{k=0}^{\infty} \frac{y^{k}}{k!} \...
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Computing Hodge numbers by point counting

In the lecture note of Bhatt from Arizona winter school 2017, there is an exercise which claims if X is a proper smooth scheme defined over $\mathbb{Z}[1/N]$ and if there is a polynomial $P$ such that ...
ali's user avatar
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Riemann hypothesis for exponential sum

Recently I've heard about the Riemann hypothesis for one-variable exponential sums, which states as For a polynomial $f\in\mathbb{F}_{p^k}[x]$ of degree $d$ and a character $\chi$ of $(\mathbb{F}_{p^...
Jugendtraum's user avatar
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For which number fields we know the nonexistence of Stark zeros?

Let $L$ be a number field and let $\zeta_L(s)$ be its associated Dedekind zeta function. It is known that $\zeta_L(s)$ has at most one zero in the region $$1 - \frac1{4 \log d_L} \leq \sigma \leq 1, \...
Molan's user avatar
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Regularity properties of Minakshisundaram–Pleijel zeta function

Let $(M,g)$ be a closed (compact, no boundary) smooth $n$-dimensional Riemannian manifold. The Laplace–Beltrami operator $\Delta_g$ on $M$ has discrete spectrum $(\lambda_j)_j$ (indexed without ...
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Ihara zeta function and closed paths and trails

Let $\Gamma$ be a finite graph. There seem to be two definitions of closed path in the literature. In one, a closed path is just a walk whose starting vertex is the same as the ending vertex. (Let us ...
H A Helfgott's user avatar
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Honda-Tate theorem and prescribing roots of $L$-functions

I'm currently working with Artin $L$-functions in function field extensions (on one variable, over finite fields). I have heard about a theorem of Honda-Tate which vaguely states that a polynomial $P \...
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Deformations of the Riemann zeta function

Consider the Dirichlet series (for fixed $0 < a \leq 1$): $$\zeta_a(s) = \sum_{n\geq 1}\frac{a^n}{n^s}$$ which reduces to the Riemann zeta function for $a=1$. What is known about this function, ...
Asvin's user avatar
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Are there any extensive treatments on rational zeta series?

I've been trying to find an extensive, in-depth treatment of rational zeta series. Via the Wikipedia article on the topic, I've found two articles on this subject. While they are certainly very ...
Max Muller's user avatar
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On $L$-function of permutation representation

I came across the statement in a book: Let $k$ be a number field and $K$ be a Galois extension of $\mathbb Q$ containing $k$, with Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ and let $G_k:=\...
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Less fundamental applications of Zeta regularization:

As we all know, zeta regularization is used in Quantum field theory and calculations regarding the Casimir effect. Are there less fundamental applications of zeta function regularization? By "less ...
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On existence of conjecture relating prime zeta function:

There is an article on Wikipedia about prime zeta function (PZF): https://en.m.wikipedia.org/wiki/Prime_zeta_function In that article , there is table of fairly accurate values of PZF for different ...
bambi's user avatar
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Regularizing the sum of all primes

In the spirit of a similar question for the harmonic series, is there a way to regularize the (divergent) sum of all primes? $$ \sum_{p \text{ prime}} p $$ Neither of these questions obtained a ...
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What's the meaning of the nontrivial zeros of Selberg zeta function?

In the case of arithmetic variety over finite field, the zero points of the Hasse-Weil zeta function reflect the pure weights (i.e. dimension). On the other hand, in the case of the Selberg zeta ...
elliptic curve's user avatar
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Could a motivic spectrum have a "zeta function"?

I'm currently learning about zeta functions, so I apologize in advance if this is riddled with nonsense. Suppose you have a sequence $E=(E_0,E_1,...)$ of motivic spaces along with structure maps $s_i:\...
Stephen McKean's user avatar
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Characterization of turning points for the Ramanujan's zeta function in the spirit of a definition by Arias de Reyna and van de Lune

In [1] the authors provided a definition and characterization of turning points for the Riemann's zeta function. In this post I denote the Ramanujan's zeta function as $$\varphi(s)=\sum_{n=1}^\infty\...
user142929's user avatar
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A principle around the Ramanujan's zeta function in short intervals

Here $s=\sigma+it$ denotes the complex variable. We denote the Ramanujan's zeta function $$\varphi(s)=\sum_{n=1}^\infty\frac{\tau(n)}{n^s}$$ for $\Re(s)>7$, where $\tau(n)$ is the Ramanujan tau ...
user142929's user avatar
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Zeros of partial sums of the Ramanujan's zeta function

In this post we consider the Ramanujan tau function $\tau(n)$, see the Wikipedia Ramanujan tau function, and we consider partial sums of its corresponding Dirichlet series (see for example the article ...
user142929's user avatar
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Prove duality of multiple zeta values by Extended Double Shuffle Relations

It is easy to prove the duality theorem of multiple zeta values (MZVs) by the integral representation of MZVs. However, how does one prove MZV's duality theorem solely by Finite Double Shuffle ...
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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 ...
Julian Chaidez's user avatar
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A generalization of gamma function

For $\alpha\in\mathbb{C}$, I defined the "complex-weighted" Hurwitz zeta function \begin{eqnarray*}\displaystyle \zeta^{\alpha}(s,w)=\frac{1}{\Gamma(s)}\int_0^{\infty} \frac{e^{-wt}}{(1-e^{-t})^{\...
days_of_wild's user avatar
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The local zeta-functions of some cubic plane curves

This question is motivated by the work presented in article 358 of Gauss' Disquisitiones Arithmeticae. For the sake of completeness, let me say something about the background and present the question ...
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Implementing zeta functions of algebraic varieties in SAGE

I am fairly new to sage, I was studying zeta functions of hypersurfaces over finite fields and I don't know how to compute them in Sage. Are there any packages that do most of the work, or maybe some ...
Martin Ortiz's user avatar
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Books on complex analysis for self learning that includes the Riemann zeta function?

I am searching for an introductory book in the field of complex analysis for self learning, that would contain the following: Analytic number theory : the connection between complex analysis and ...
user144435's user avatar
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Can the corollary of the Ihara–Bass formula be extended to $ u^2 = 1 $?

Suppose there is a finite undirected graph $G(V,E)$ having $n$ vertices and $m$ edges. The non-backtracking matrix $B$ is indexed by $2m$ directed edges and defined as $$ B(a \to b, c \to d) = \delta_{...
Eli4ph's user avatar
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What is the current fastest method to calculate Lerch's Phi transcendent?

Lerch's Phi transcendent is $$ \Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(k+a)^s} $$ I am trying to compute this for the following parameters: $z$ is complex, $|z| \approx 1$ and $|z|$ < 1 (...
Mike Battaglia's user avatar
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1 answer
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How to measure how much a rational function/a singularity of variety is complicated?

There are some theorems about various zeta functions which states the rationality of those. For example, when you consider Igusa's zeta function, roughly the generating series of solutions mod $p^n$ ...
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Functional Equation of Zeta Function on Statistical Model

I've been studying [1] because I was interested in his ideas on the zeta function. I'll define it here (c.f. p. 31): The Kullback-Leibler distance is defined as $$ K(w)=\int q(x)f(x, w)dx\quad f(x,w)...
Matt Cuffaro's user avatar
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1 answer
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Igusa zeta functions of univariate polynomials: $\mathbb{Z}_p$ or $\mathbb{Q}_p$ in this statement

Let $f\in\mathbb{Z}_p[X]$ and let $Z_{f,p}(T)\in\mathbb{Z}_{(p)}(T)$ be the $p$-adic Igusa zeta polynomial (i.e. $Z_{f,p}(p^{-s})$ is the $p$-adic Igusa zeta function in the complex variable $s$, with ...
Maurizio Moreschi's user avatar
8 votes
1 answer
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Riemann Hypothesis, Primes and Groups

Let $G$ be a finite group $S\subset G$ a generating set $|g|=$ word length with respect to $S$. Set $$ \sigma(G) = \sum_{H \le G} [G:H]$$ Let $\rho$ be the regular representation and set $A_G := \...
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7 votes
1 answer
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Word length zeta function

Let $G$ be a group with a finite symmetric set $S$ of generators. Let $\ell_S(x)$ denote the word-length of a given $x\in G$. For $s\in\mathbb C$ set $$ Z(s)=\sum_{x\in G^*}\ell_S(x)^{-s}, $$ where $G^...
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5 votes
1 answer
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Results in an article by Siegel

Studying the Eisenstein cocycle by Sczech, I noticed that to understand its connection with the values at negative integers with zeta functions it is necessary to understand the resuts by Siegel in ...
efs's user avatar
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3 answers
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Motivation for zeta function of an algebraic variety

If $p$ is a prime then the zeta function for an algebraic curve $V$ over $\mathbb{F}_p$ is defined to be $$\zeta_{V,p}(s) := \exp\left(\sum_{m\geq 1} \frac{N_m}{m}(p^{-s})^m\right). $$ where $N_m$ is ...
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1 answer
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Is there research on the special values of the zeta function outside the integers?

This question quotes from this article, but I've noticed this pattern in the literature I've read. "The values or better the leading coefficients at integral arguments of the L-functions of ...
Matt Cuffaro's user avatar
3 votes
1 answer
170 views

Bounding the second moment of $|\zeta(\sigma+i t)|^2$ for $0<\sigma<1$

Let $$I(\sigma,T)=\int_0^T |\zeta(\sigma+ i t)|^2 dt.$$ Unconditional bounds and asymptotics for $I(\sigma,T)$, $1/2\leq \sigma <1$, have been known since Hardy and Littlewood (see Chapter 7 of ...
H A Helfgott's user avatar
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14 votes
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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 ...
wonderich's user avatar
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7 votes
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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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25 votes
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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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17 votes
2 answers
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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 ...
Timothy Chow's user avatar
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1 answer
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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 ...
mjungmath's user avatar
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1 answer
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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 $...
Nico A's user avatar
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1 answer
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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}...
Junhyeong Kim's user avatar
2 votes
1 answer
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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 ...
Maurizio Moreschi's user avatar
3 votes
1 answer
320 views

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 ...
Maurizio Moreschi's user avatar
3 votes
0 answers
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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 ...
Steve Huntsman's user avatar
2 votes
2 answers
170 views

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 ...
Elle Najt's user avatar
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4 votes
1 answer
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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 ...
Asvin's user avatar
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14 votes
2 answers
969 views

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/...
4 votes
0 answers
294 views

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. ...
user39115's user avatar
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