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2
votes
1answer
137 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 ...
3
votes
0answers
88 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
0answers
87 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 ...
3
votes
2answers
207 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}} ...
0
votes
0answers
51 views

Convergence of Selberg type Zeta function

Let $X=G/K$ be a Riemannian symmetric space without compact factor. We can assume $G$ is connected and real reductive. $K$ is a maximal subgroup of $G$. Let $\Gamma$ be a discrete torsion free ...
1
vote
0answers
61 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
0answers
72 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
1answer
124 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 ...
0
votes
0answers
41 views

Some of the real zeros of those $k^{th}$ derivatives are also simple?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the ...
9
votes
1answer
439 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 ...
0
votes
0answers
40 views

Can we deduce something about the nature of those solutions?

To describe the problem, we note that we can find an affine model for any elliptic curve $C$ over $ℚ$ in Weierstrass form $$C:y^2=x^3+ax+b$$ with $a,b∈ℤ$. The full L-series of $C$ is given by ...
3
votes
2answers
348 views

Local factors of Hasse-Weil zeta function - what do they have in common?

Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of ...
8
votes
2answers
453 views

is there a p-adic Borel theorem?

Let $F$ be a number field. Denote, as usual, $\mathcal{O}_F$ the ring of integers and $r_1$, $r_2$ the number of real and complex embeddings. Let $\zeta_F(s)$ be the Dedekind zeta function of $F$. ...
1
vote
1answer
143 views

Real points $a∈ℝ$ such that the equation $f^{(k)}(s)=a$ have a finite number of real solutions $s$ for some $k$

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the ...
3
votes
0answers
110 views

Weierstrass's elliptic function-type zeta function

What is known about the following Weierstrass's elliptic function-type zeta function $\sum_{m,n \in \mathbb{Z}} \frac{1}{(z+m+n\tau)^s}$, for $z \in \mathbb{C} \backslash \mathbb{Z} + \tau ...
2
votes
1answer
110 views

Alternating series $\sum_{k=1}^{\infty}(-1)^{k+1} (H_k)^p z^k$ and multiple zeta values

Motivated by analytic continuation of solutions of a Picard-Fuchs equation, we encountered sums of the following form $S(z;p)=\sum_{k=1}^{\infty}(-1)^{k+1} (H_k)^p z^k$ where $H_k = \sum_{n=1}^{k} ...
1
vote
2answers
253 views

prime zeta function when $0<s<1$ [closed]

I will not be surprised if this question seems trivial in MO but i asked it first in MathSE and i did not get an answer. So, here it is: I would like to know if there is a good estimate for the sum ...
1
vote
0answers
56 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 ...
1
vote
2answers
191 views

zeta-function regularized integrals

I gather that the following two identities about $\xi(3)$ hold via some notion of zeta-function regularized integrals. $\xi(3) = \frac{(2\pi)^3}{3}\int _0 ^\infty d\lambda \frac{\sqrt{\lambda} }{1 + ...
2
votes
1answer
263 views

Epstein zeta functions

Does anyone know an expression (in terms of simpler functions) for the following Epstein Zeta function: $\sum \frac{1}{(m^2+m n+n^2)^s}$ I know an expression (in terms of the Dirichlet Beta ...
2
votes
0answers
188 views

Automorphicity of L-Factors of Zeta Functions

Associated to a variety over a number field $K$, one has a family of ``Hasse--Weil'' L-functions, which can be combined (as an alternating product) to give the Hasse--Weil zeta function of the ...
10
votes
0answers
236 views

L-Functions of Varieties, Zeta Functions of Their Models

Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the ...
6
votes
1answer
245 views

Gamma Factors for Zeta Functions of Abelian Varieties

If $X$ is a scheme of finite type over $\mathbb{Z}$, and $X_0$ denotes its set of closed points, then one can define its zeta function on the half plane $Re(s)>\text{dim}(X)$: \begin{equation} ...
1
vote
0answers
249 views

Does the property (P) holds true for the derivatives of $L$?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. As $s$ takes on real negative values, there are trivial zeros ...
1
vote
1answer
558 views

An integral representation of the Riemann zeta function

I am referring to the equality in equation $3.29$ (page 12) and $4.20$ (page 17) in this paper. I am unable to recognize where this comes from or what is the general expression for values other than ...
4
votes
1answer
209 views

Relation between Lee and Yang' s “circle theorem”, zeta functions and Weil conjectures?

Ruelle mentions ( http://www.ihes.fr/~ruelle/PUBLICATIONS/%5B94%5D.pdf ) Lee and Yang' s "circle theorem", which comes from statistical mechanics and shall have not yet explored connections with zeta ...
4
votes
0answers
240 views

Zeta functions for infinite abelian extensions?

Are there any reasonable ways to define zeta function of a maximal abelian extension $\mathbb{Q}^{ab}$ of the rationals, or at least zeta functions of some its infinite-dimensional subfields? In ...
2
votes
0answers
202 views

computing a certain contour integral [closed]

I want to compute an integral along a vertical line segment. The function I'm integrating involves the zeta-function, and usually the way such integrals are done treats the line segment as one side ...
4
votes
0answers
308 views

The Riemann Zeta Function summing over the Gamma Function

Has anyone studied a function of the form $$\eta(s) = \sum_{n=1}^{\infty} \frac{1}{\Gamma(n)^{s}} = \sum_{n=0}^{\infty}\frac{1}{k!^s}$$ This series is appearing in my research on the volumetric ...
6
votes
6answers
857 views

Sequences equidistributed modulo 1

Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results. H. Weyl (1909): The fractional part of the sequence $\alpha n$ is equidistributed modulo 1. ...
0
votes
1answer
148 views

Can we find a set of elliptic curves over rationals associated with $f$?.

We know that for each elliptic curve over rationals, we can define the Dirichlet series of the Hasse–Weil $L$-function, i.e., the function associated with an elliptic curve over rationals. Then my ...
1
vote
3answers
648 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 ...
6
votes
1answer
361 views

Efficient (divergent) summation for sum of zetas at negative arguments?

In a question in MSE (see bottom of my own answer) I'm considering the following series, depending on a parameter m: $$ L(m) = -\zeta(1m)/1 - \zeta(2m)/2 - \zeta(3m)/3 - \ldots $$ where I want to make ...
47
votes
4answers
3k views

Are there refuted analogues of the Riemann hypothesis?

The classical Riemann Hypothesis has famous analogues for function fields and finite fields which have been proved. It has by now very many analogues, many of them still open. Are there important ...
0
votes
1answer
161 views

This might be a trivial question on Hurwitz's zeta function.

In the book I am reading they write that for Hurwitz zeta function, $\zeta(x,s)=\sum_{n=0}^{\infty} \frac{1}{(x+n)^s}$, the next sum in the RHS converges for $\Re(s)>-1$, and I don't see how ...
2
votes
1answer
216 views

The Hasse-Weil L-function and some equations

Let $f$ be an analytic function verfifying $f(s)=\epsilon f(2-s)$ where $\epsilon=\pm 1$. The expression of Hasse-Weil L-function $f$ is ...
18
votes
4answers
1k 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 "expectional zeros" of course first ...
5
votes
2answers
647 views

Is there an explicit expression for the imaginary part of some non-trivial zero of zeta,

Is there an explicit expression for the imaginary part of some non-trivial zero of zeta, in terms of well-known constants, such as say $\gamma$ or $\pi$ say ?
1
vote
0answers
75 views

binomial transform, Hurwitz zeta function

For $j,n\in\mathbb Z_+$, let $$ L_{j,n}^{(t)}= \sum_{m=0}^{n} \Bigl(-\frac 12\Bigr)^{n-m}{n\choose m}{m+j+1\choose m+1} \left( \frac {1}{t+\frac 12}\right)^{m+j+2} $$ and $$ L_{j,n} ...
0
votes
2answers
225 views

Ihara zeta function (graph theory) coefficients using a line graph [closed]

I'VE COMPLETELY REVISED MY QUESTION I wish to take a simple undirected graph (i.e. the complete graph K_4) Arbitrarily direct said graph, and then create a line graph from the directed version of ...
0
votes
0answers
191 views

Analytical continuation of the reciprocal of the Zeta function

Is the reciprocal of the Zeta function analytically continuable? As $1/\zeta(n) = \sum_{n=1}^\infty \mu(n)/{n^s}$, this does not look obvious.
0
votes
1answer
113 views

Are potential complex zeros not on the critical line of Dedekind zeta function in quadruples?

This question arose from sums over zeros of Dedekind zeta function. It is known that complex zeros of Dedekind zeta function are in pairs $\rho, 1 - \rho$. Is it true that potential complex ...
4
votes
0answers
303 views

Does the Riemann hypothesis for liftable varieties over a finite field imply the Riemann hypothesis for all varieties over a finite field

The Riemann hypothesis for varieties over a finite field has been proven by Deligne. Still I would like to ask the following question. A variety $X$ over a finite field $k$ is liftable if there ...
1
vote
0answers
136 views

Special values of zeta functions and extensions of base fields.

Let $X$ be a scheme of finite type over a finite field $k=\mathbb{F}_{q}$ of $q$ elements. Then, one can define the zeta function $Z_{X/k}(T)$ of $X$ ovet $k$ as $\prod_{x\in ...
0
votes
2answers
295 views

The origin of the root number $w(C/ℚ)=±1$ (the sign of the functional equation)

The motivation for this question is the same as in my previous question in MO: http://mathoverflow.net/questions/115179/real-root-1-of-the-hasse-weil-l-function-of-c-over I am just curious to know ...
0
votes
2answers
393 views

Hadamard's product formula for the derivative

Let $f$ be an entire function of order $ρ<\infty$. Assume that $f$ does not vanish identically on $\mathbb{C}$. Then, we know that $f$ has a Hadamard's product formula $$ f(s) =e^{g(s)}s^{r}\prod ...
1
vote
1answer
239 views

Question about the function $\zeta(s) \pm \dfrac{1}{\zeta(1-s)}$

It is easy to see that the function: $$\zeta(s) \pm \dfrac{1}{\zeta(1-s)}$$ has a pole at each non-trivial zero $s=\rho_n$. However, after some experiments with this function, I would like to ...
-10
votes
2answers
705 views

Riemann Siegel function and gamma function

I ask about an idea to prove this formula: $Γ(1/2-iβ)=((√π)/(√(coshπβ)))exp(-i(2ϑ(β)+βln2π+arctan(tanh(1/2)πβ)))$ where $ϑ(β)$ is the Riemann Siegel function.
6
votes
1answer
987 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 ...
11
votes
1answer
318 views

Sets with zeta functions that are not the primes

Does there exist a set $S \subset \mathbb N$ such that the Dirchlet density of $S$ is well-defined and positive, the Dirchlet density of $S \cap \operatorname{PRIMES}$ is well-defined and zero, and: ...