**4**

votes

**1**answer

256 views

### 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 ...

**0**

votes

**2**answers

280 views

### The Zeta Function Before Riemann [duplicate]

Leonhard Euler studied the function that is now known as the Riemann zeta function. I have not found the notation $\zeta$ in any of the works of any mathematicians prior to Bernhard Riemann's paper On ...

**11**

votes

**0**answers

174 views

### Weil conjectures for higher dimensional cycles?

Let $X$ be a smooth projective variety over $\mathbb{F}_{q}$. For each pair of positive integers $n$ and $d$, let $\text{Chow}_{n,d}(X)$ denote the (coarse) moduli space of $n$-cycles of degree $d$ on ...

**1**

vote

**1**answer

153 views

### What does the higher coefficients of ihara zeta function reveal?

Assume we have a graph $G=(V,E)$.
The ihara zeta function $Z(G,u)$ is of form $$\frac1{\displaystyle\sum_{i=0}^{2|E|}c_iu^i}$$
A graph which has $|E|$ edges cannot have a simple cycle of length ...

**2**

votes

**1**answer

208 views

### 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?

**11**

votes

**1**answer

440 views

### Multiple zeta values at negative integers

For a tuple of integers $\underline{s}:=(s_1,\dots, s_d)$, the multiple zeta value (MZV) at $\underline{s}$ is defined as:
$$ \zeta(s_1,\dots, s_d):= \sum_{n_1>\dots>n_d>0}\frac 1 {n_1^{s_1}\...

**14**

votes

**1**answer

477 views

### Growth of $\zeta_{\mathbf Q[\cos(\frac{\pi}{2^{n+1}})]}(2)$

Let $K_n$ be the field $\mathbf Q[\cos(\frac{\pi}{2^{n+1}})]$ (the real subfield of the cyclotomic field $\mathbf Q[e^{\frac{i\pi}{2^{n+1}}}]$).
Is there anything known about the growth of the ...

**16**

votes

**2**answers

853 views

### Can the Dedekind zeta function distinguish between real and imaginary quadratic number fields?

Suppose I am given a machine that gives me the coefficients $a_1$, $a_2$, $a_3$, ... of a Dirichlet series
$$\sum_1^{\infty} \frac{a_n}{n^s} $$
and assume that I know that this Dirichlet series is the ...

**33**

votes

**2**answers

2k views

### Is there a “quantum” Riemann zeta function?

Occasionally I find myself in a situation where a naive, non-rigorous computation leads me to a divergent sum, like $\sum_{n=1}^\infty n$. In times like these, a standard approach is to guess the ...

**17**

votes

**1**answer

396 views

### Convergence of zeta functions for schemes of finite type over the integers

In his lecture "Zeta functions and $L$-functions", Serre presents a very elegant proof of the convergence of the zeta function
$ \zeta (X,s) = \prod_{x \in |X|} (1- N(x)^{-s})^{-1}$ in the half plane ...

**32**

votes

**3**answers

4k views

### The Hardy Z-function and failure of the Riemann hypothesis

David Feldman asked whether it would be reasonable for the Riemann hypothesis to be false, but for the Riemann zeta function to only have finitely many zeros off the critical line. I very rashly ...

**27**

votes

**0**answers

511 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 ...

**4**

votes

**1**answer

104 views

### Calculation of one constant similar to MZV

The series arose in the calculation of Mean value of a function associated with continued fractions:
$$C=\sum_{1\le b\le d<\infty}\frac{1}{b(b+d)d^2}.$$
Obviously
$C=C_1-C_2,$
where
$$C_1=\sum_{1\...

**3**

votes

**2**answers

85 views

### Zeros of the series $\phi(s, a) = \sum_{n \geq 0} e^{-an}(n + 1)^{-s}$

I have been recently interested in the series $\phi(s, a)$ of the title.
There, a is defined to be any positive real number and s is a complex variable.
The main reason for my curiosity is that the ...

**8**

votes

**0**answers

206 views

### Arithmetic zeta function and local zeta functions

For the arithmetic zeta function of (say) a nonsingular projective variety $X$, one has the following Euler product
\begin{equation}
\zeta_X(s) = \prod_{p\ \mbox{prime}}\zeta_{X\vert\mathbb{F}_p}(s),
...

**21**

votes

**4**answers

2k views

### Good uses of Siegel zeros?

The short version of my question goes: What is known to follow from the existence of Siegel zeros?
A longer version to give an idea of what I have in mind: The "exceptional zeros" of course first ...

**2**

votes

**1**answer

857 views

### Is the integral always nonzero?

Let
$$I_{n,p,a}:=\int_0^{\infty-} \frac{g_{n,a}(t)}{t^{p+1}} \, dt,$$
where
$$(*)\qquad\qquad\qquad n\in\mathbb N,\quad -\infty<a<\infty,\quad p_{n,a} < p < n,\qquad\qquad\qquad\qquad\...

**2**

votes

**0**answers

36 views

### Local zetafunction of T-group and Lie-ring coincide

Grunewald, Segal and Smith define in "Subgroups of finite index in nilpotent groups" a zetafunction associated to a finitely generated, torsion-free nilpotent group G by counting normal subgroups ...

**45**

votes

**8**answers

7k views

### Why does the Gamma-function complete the Riemann Zeta function?

Defining $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ yields $\xi(s) = \xi(1 - s)$ (where $\zeta$ is the Riemann Zeta function).
Is there any conceptual explanation - or ...

**-7**

votes

**1**answer

1k views

### Non-standard numbers and exponential form of Zeta function [closed]

Basic idea
For a long time I was looking for a numerical system that would allow to compare infinite sets. In contrast to Cantor's approach that empathizes the possibility of on-to-one correspondence ...

**8**

votes

**2**answers

323 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) ...

**28**

votes

**4**answers

4k views

### What is Ricardo Pérez-Marco's eñe product? Does it explain his statistical results on differences of zeta zeros?

The number theory community here at University of Michigan is abuzz with talk of this paper recently posted to the arxiv. If you haven't seen it already, the punch line is that the global differences ...

**2**

votes

**0**answers

38 views

### Computations of some character sums/zeta function

I'm currently trying to "compute" the zeta function of some hyperelliptic curves over a finite field (of odd characteristic). Precisely, let $b\in\mathbb{F}_q^\times$ be a non zero element (I also ...

**7**

votes

**0**answers

274 views

### Asymptotics of A261668

In Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras, Proposition 10.8, Jianqiang Zhao mentiones the sequence:
$$a_n=\sum_{...

**17**

votes

**1**answer

554 views

### stable homotopy groups and zeta function

I have heard during a discussion that there is a well known relation between the stable homotopy groups of a sphere (more precisely the order of stable homotopy groups of localized sphere spectrum ...

**6**

votes

**1**answer

300 views

### Complex Geometry Consequesnces of Serre's Kahler-Zeta Function

Serre's famous paper Analogues K\"ahl\'eriens de Certaines Conjectures de Weil proves an analogue of the Weil conjectures for compact K\"ahler manifolds. It would go on to inspire the line of attack ...

**6**

votes

**1**answer

162 views

### Zeta-Determinant for shifted Laplacians on the circle

Consider on the circle $S^1$ the operator
$$L := - \frac{\partial^2}{\partial \theta^2} + c$$
for some constant $c \in \mathbb{R}$.
What is its $\zeta$-regularized determinant?
This should be well-...

**3**

votes

**0**answers

827 views

### Proof that derivative of Hurwitz Zeta by the first argument is not expressable in terms of Hurwitz Zeta

The set of elementary functions is defined so that it to be closed against operation of differentiation. It is also evidently close against discrete differentiation.
In the discrete calculus there is ...

**0**

votes

**0**answers

87 views

### Bound of Chebyshev function and zeros of zeta function

It is an elementary argument (such as in Multiplicative Number Theory, section 18) that, if the Chebyshev's function $f(x) = \sum_{n \le x} \Lambda(x) = x + O(x^\alpha)$ for some $\alpha < 1$, then ...

**11**

votes

**2**answers

1k views

### Analogues of the Riemann-Roch Theorem

In his thesis, Tate derives a Poisson formula on the adeles and a theorem which he calls the "Riemann-Roch Theorem". More specifically, given a continuous, $L^1$ function $f$ on the adeles such that ...

**7**

votes

**1**answer

1k 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 ...

**6**

votes

**2**answers

951 views

### A (likely) positivity property of the Lerch zeta-function

The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where
$$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$
is the Lerch zeta-...

**20**

votes

**6**answers

5k 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 ususally the books start with, e.g. tricky change of variable of Gamma function or other seemly unmotivated ...

**11**

votes

**2**answers

347 views

### Tauberian theorem with better error term

This is a fairly vague question.
Suppose we have a sequence of positive numbers $(c_n)_n$ and we want to find an asymptotic formula for $S(x) = \sum_{n \leq X} c_n$. In favorable circumstances, ...

**4**

votes

**0**answers

198 views

### Equivalence of Euler products of Dirichlet series and Meromorphic continuation

Suppose $(f_n(s))_n$ and $(g_n(s))_n$ are two sequences of Dirichlet series with positive coefficients such that $\exists \alpha\in\mathbb R$ such that for all $s\in\mathbb C$ with $\Re(s)>\alpha$ ...

**1**

vote

**1**answer

157 views

### Hasse-Weil L-Functions of CM Abelian Varieties

In Shimura's paper "On the Zeta Function of an Abelian Variety With Complex Multiplication", in his terminology, the `one-dimensional part' of the zeta function is identified with a Hecke $L$-function ...

**2**

votes

**1**answer

266 views

### Explicit examples of Hasse--Weil zeta-function calculations for curves

The problem of calculating Hasse--Weil zeta-function for a given curve $C/\mathbb{F}_p$ over a finite field is far from being easy, especially for large genus (as discussed by Wouter Castryck at https:...

**-9**

votes

**2**answers

905 views

### Riemann Siegel function and gamma function

I ask about an idea to prove this formula:
$Γ(1/2-iβ)=((\sqrt{π})/(\sqrt{\coshπβ}))\exp(-i(2ϑ(β)+βln2π+\arctan(\tanh(1/2)πβ)))$
where $ϑ(β)$ is the Riemann Siegel function.

**4**

votes

**1**answer

138 views

### Functional equations for spectral zeta function

Functional equation in the theory of zeta functions is one of the important components of this theory.
I am interested to know whether the similar property, having functional equation, for the ...

**27**

votes

**2**answers

2k views

### The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.

I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt ...

**2**

votes

**1**answer

230 views

### Special values of Hecke L-function

The Dedekind zeta function for a number field $K$ is defined as
$\zeta_K(s)=\sum_{I\subset O_K} (N_{K/\mathbb{Q}}(I))^{-s}$.
By attaching a Hecke character $\psi$, we can define $L(s,\psi)=\sum_{I\...

**3**

votes

**0**answers

115 views

### Zeta functions with Brauer class

In algebraic geometry there are examples when a variety $X$ is somehow related (I call it double-mirror) to another variety $Y$, together with
a 2-torsion Brauer class. By "related" I mean statements ...

**2**

votes

**0**answers

114 views

### Orders of Clifford algebra

Let $C_n$ be the Clifford algebra over $\mathbb{Q}$ associated to negative definite quadratic form $-I_n$ (i.e. $-x_1^2-\dots-x_n^2$). Let $\mathcal{O}$ be a $\mathbb{Z}$-order of $C_n$.
Q1) Is it ...

**6**

votes

**2**answers

2k views

### What is the Beilinson regulator?

Trying to understand answer to this question.
What is the (Beilinson) higher regulator of a number field?

**1**

vote

**3**answers

759 views

### A generalisation of the Birch and Swinnerton-Dyer conjecture

We know that the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and Generalized Riemann hypothesis. My question is about the existence of a similar generalisation of the Birch ...

**4**

votes

**2**answers

279 views

### Inequality for an integral involving $ \exp $, $ \sin $ and $ \cos $

Let $ t > 0 $ and $ k \in \{ 0,1,2,\ldots \} $. Does the following inequality hold?
$$
\int_{k + 1/2}^{k + 3/2}
\frac{x \sin(2 \pi x)}{1 + 2 e^{2 \pi t} \cos(2 \pi x) + e^{4 \pi t}}
...

**1**

vote

**0**answers

67 views

### density of zeroes of Epstein zeta functions on vertical strips

There are many results (e.g. Davenport and Heilbronn) asserting that Epstein zeta functions in general have zeroes outside the line $\mathrm{Re\ } s = n/4$. Is there any result about the density of ...

**0**

votes

**0**answers

92 views

### Are the first ten zeros of this Dedekind zeta function non-simple?

This question
asks about the zeros of the zeta function of the number field with defining
polynomial:
...

**1**

vote

**1**answer

143 views

### Factorisation of local quaternionic zeta functions

Vignéras, in her Arithmetics of quaternion algebras, begin chapter II.4 recalling that we know the number of integer ideals of fixed norm of a quaternion algebra $H$ over a local field $K$, ramified ...

**10**

votes

**1**answer

497 views

### Is it easy to prove that $\sum_n |X(\mathbb{F}_{q^n})| t^n$ is rational?

Background: Let $X$ be an algebraic variety over a finite field $\mathbb{F}_q$. One of the successes of Etale cohomology - previously achieved by Dwork- was proving the rationality of the Zeta ...