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.

Filter by
Sorted by
Tagged with
11 votes
1 answer
796 views

Moduli space of germs of riemannian metrics

Let $S$ be the set of germs of riemannian metrics near $0$ on $\mathbb R^n$. It is acted on by the group $\textrm{Diff}$ of germs of diffeomorphisms of $\mathbb R^n$ preserving $0$. Let's denote by $S^...
7 votes
1 answer
1k views

Contemporary introduction to Godement-Jacquet "Zeta functions of simple algebras"

The question is in the title: The book Godement-Jacquet "Zeta functions of simple algebras" is from 1971. Has there ever been a textbook introduction to this material, or at least part of it?...
3 votes
0 answers
199 views

Schemes with common zeta function

If $S_\zeta$ is the set of all separated schemes of finite type over $\mathbb{Z}$ that have the same arithmetic zeta function $\zeta$, what more can we say about $S_\zeta$ assuming it is non-empty?
7 votes
0 answers
637 views

Update on "A Mad day's work" by Cartier

In his paper "A mad day's work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry," Cartier discusses in the last sections 8 and 9 the role of ...
2 votes
0 answers
584 views

Summation form of the Hasse-Weil zeta function?

The only way I know to write the Hasse-Weil zeta function of an elliptic curve is as a product over the local zeta factors which are rational functions. To me, this appears like an Euler product. Is ...
4 votes
0 answers
141 views

Riemann-Siegel formula for Dirichlet characters

After unearthing and giving a proof of what is now known as the Riemann--Siegel formula for the Riemann zeta function enabling the computation of $\zeta(1/2+iT)$ in time $O(T^{1/2})$, in 1943 Siegel ...
3 votes
2 answers
488 views

Dedekind Zeta functions of Biquadratic fields

Let $F/ \mathbb{Q}$ be a biquadratic field of Galois group $C_2 \times C_2$. Then I know that the Dedekind Zeta function of $F$ can be factored into $L$-functions as; $$\zeta_F(s) = \zeta(s) L(s, \...
22 votes
1 answer
830 views

Why is the regulator of the degree 11 extension of $\mathbb{Q}$ so large?

Let $K_\infty/\mathbb{Q}$ denote the $\hat{\mathbb{Z}}$-extension of $\mathbb{Q}$. Then for each $n\geq1$, $K_\infty$ has a unique subfield $K_n$ of degree $n$ over $\mathbb{Q}$. The fields $K_n$ are ...
8 votes
1 answer
1k views

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 ...
28 votes
1 answer
919 views

Relation between Schanuel's theorem and class number equation

(Crossposted on math stack exchange: https://math.stackexchange.com/questions/4040249/relation-between-schanuels-theorem-and-class-number-equation) It was recently brought to my attention that there ...
74 votes
4 answers
9k views

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 ...
37 votes
8 answers
11k views

How does one motivate the analytic continuation of the Riemann zeta function?

I saw the functional equation and its proof for the Riemann zeta function many times, but usually the books start with, e.g. tricky change of variable of Gamma function or other seemingly unmotivated ...
3 votes
0 answers
482 views

Serge Lang's proof of Brauer-Siegel theorem

I was reading through chapter 16 of Lang's Algebraic Number Theory book. The chapter is fully devoted to proving the Brauer-Siegel theorem: Let ${(k_n/ \mathbb{Q})}_n$ be a sequence of galois ...
2 votes
1 answer
244 views

The zeta regularization of $\prod_{m=-\infty}^\infty (km+u)$

Background: I'm facing the computation of the zeta regularization of the infinite product given by $$\prod_{m=-\infty}^\infty (km+u)$$ for a real positive $k$ and $\Im(u)\neq 0$. From J. R. Quine, S. ...
2 votes
0 answers
56 views

Class of differentiated Gamma functions: are there any algebras where they are elementary?

There is a class of differentiated gamma functions (DGFs) that basically can be expressed via derivatives of Gamma function. They include the Gamma function, Polygamma function, and Hurwitz Zeta ...
2 votes
0 answers
97 views

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^...
12 votes
3 answers
2k views

What is the value of $p$-adic $\zeta$-function at positive integer point?

$p$-adic zeta function is a $p$-adic interpolation of the Riemann $\zeta$-function for the values $\zeta(1−k)$, $k\ge 1$ (see $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz) ...
7 votes
4 answers
491 views

Is this closed-form summation a special case of known Lerch zeta function formulas?

With some Poisson summation manipulations (credit: Michał Pacholski) I have convinced myself of a closed form expression for this conditionally convergent series: $$\sum_{n=-\infty}^\infty \frac{e^{in\...
6 votes
0 answers
3k views

Value of $\zeta(3/2)$?

Is anything known about the value of $$\zeta(3/2)=\sum_{n\geq 1}\frac{1}{n^{3/2}}?$$ It is a classical result that $\displaystyle \zeta(2)= \frac{\pi^2}{6}$ and $\zeta(3)$ has been shown to be ...
3 votes
0 answers
101 views

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!} \...
11 votes
1 answer
2k views

Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta ...
7 votes
2 answers
905 views

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 ...
6 votes
0 answers
238 views

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 ...
1 vote
1 answer
176 views

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, \...
8 votes
1 answer
585 views

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^...
3 votes
0 answers
434 views

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

Let's consider the following bipartite cubic planar non-simple graph $\hskip2.3in$ Looking at the orientation of the edges around the vertices, it is obvious that left and right are oriented opposite. ...
2 votes
0 answers
101 views

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 ...
9 votes
3 answers
5k views

Riemann zeta at even integers

I am talking about this in a course I am teaching, and hence am wondering: what are the various derivations of the values of Riemann zeta function at even integers? There are two incredibly cool ...
4 votes
0 answers
144 views

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 ...
4 votes
0 answers
122 views

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 \...
1 vote
0 answers
99 views

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\...
4 votes
1 answer
303 views

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, ...
0 votes
1 answer
523 views

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:=\...
7 votes
1 answer
807 views

Persistent homology of Markovian dynamical systems

Consider a dynamical system $(T,X)$ that admits a Markov partition $\mathcal{M}$ (e.g., an Anosov map), and consider the corresponding 0-1 transition matrix $A$. It is commonplace to study information ...
6 votes
2 answers
543 views

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 ...
1 vote
1 answer
143 views

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 ...
0 votes
1 answer
225 views

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 ...
1 vote
0 answers
179 views

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 ...
6 votes
1 answer
390 views

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:\...
4 votes
1 answer
345 views

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 ...
4 votes
1 answer
246 views

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 $p^n$ ...
1 vote
0 answers
82 views

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 ...
14 votes
1 answer
1k views

Is there an elegant algebraic proof of this formula for quadratic field discriminants?

Consider the Dirichlet series counting discriminants of real quadratic fields. Quadratic field discriminants are "basically" squarefree integers, so the associated Dirichlet series $\sum D^{-s}$ is "...
15 votes
1 answer
945 views

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 vote
1 answer
184 views

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})^{\...
5 votes
1 answer
387 views

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 ...
3 votes
1 answer
177 views

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 ...
33 votes
1 answer
1k views

What does this connection between Chebyshev, Ramanujan, Ihara and Riemann mean?

It all started with Chris' answer saying returning paths on cubic graphs without backtracking can be expressed by the following recursion relation: $$p_{r+1}(a) = ap_r(a)-2p_{r-1}(a)$$ $a$ is an ...
2 votes
1 answer
2k views

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 ...
4 votes
2 answers
502 views

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 (...