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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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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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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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 ...
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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 ...
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Is there a known formula for fractional derivative of cot x?
I wonder if there any established formula for fractional derivative of a function $\pi \cot (\pi x)$.
I derived the following expression:
$(\pi \cot (\pi q))^{(p)}=-\frac{\zeta'(p+1,q)+(\psi(-p)+\...
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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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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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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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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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Weil Conjectures for Grassmannians
To establish the Weil conjectures for $n$-dimensional projective space over a finite field is elementary. Does there exist a simple direct proof of the conjectures for finite field Grassmannians?
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Analytical continuation of the reciprocal of the Zeta function [closed]
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.
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Evaluating the integral $\int_0^\infty \frac{\psi(x)-x}{x^2}dx.$
Let $\psi(x)=\sum_{n\leq x} \Lambda(n)$ be the weighted prime counting function. I am trying to evaluate the integral $$\kappa:=\int_{1}^{\infty}\frac{\psi(x)-x}{x^{2}}dx$$ in several different ways. ...
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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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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 variety, ...
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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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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.)
$$(*)\quad \Gamma(\mu)\, \zeta(\mu,\nu) = \int_{0}^{1} x^{\nu-1} \,(1-x)^{-1} \Bigr(\log ...
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Sequences equidistributed modulo 1
Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results.
H. Weyl (1909): The fractional part of the sequence $\alpha n$ is equidistributed modulo 1.
I....
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Geometric and arithmetic Frobenius
I read in Serre's "Lectures on $N_X(p)$" that when $X$ is a scheme defined over $\mathbb{F}_q$ (a finite field), the geometric Frobenius $F: X \mapsto X$ is defined by fixing every element of the ...
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The Zeta Function Before Riemann [duplicate]
Leonhard Euler studied the function that is now known as the Riemann zeta function. I have not found the notation $\zeta$ in any of the works of any mathematicians prior to Bernhard Riemann's paper On ...
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Weil Conjectures Analog for Multivariate Zeta Functions
We know that the Riemann zeta function can be generalized to multivariate zeta functions.
Is there a multivariate analog of the Weil conjectures?
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Multiple zeta values at negative integers
For a tuple of integers $\underline{s}:=(s_1,\dots, s_d)$, the multiple zeta value (MZV) at $\underline{s}$ is defined as:
$$ \zeta(s_1,\dots, s_d):= \sum_{n_1>\dots>n_d>0}\frac 1 {n_1^{s_1}\...
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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 ...
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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 ...
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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 ...
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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 ...
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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
$$C_1=\sum_{1\...
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Zeros of the series $\phi(s, a) = \sum_{n \geq 0} e^{-an}(n + 1)^{-s}$
I have been recently interested in the series $\phi(s, a)$ of the title.
There, a is defined to be any positive real number and s is a complex variable.
The main reason for my curiosity is that the ...
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Arithmetic zeta function and local zeta functions
For the arithmetic zeta function of (say) a nonsingular projective variety $X$, one has the following Euler product
\begin{equation}
\zeta_X(s) = \prod_{p\ \mbox{prime}}\zeta_{X\vert\mathbb{F}_p}(s),
...
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Good uses of Siegel zeros?
The short version of my question goes: What is known to follow from the existence of Siegel zeros?
A longer version to give an idea of what I have in mind: The "exceptional zeros" of course first ...
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Is the integral always nonzero?
Let
$$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\...
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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 ...
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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 ...
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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:
$$a_n=\sum_{...
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stable homotopy groups and zeta function
I have heard during a discussion that there is a well known relation between the stable homotopy groups of a sphere (more precisely the order of stable homotopy groups of localized sphere spectrum ...
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Zeta-Determinant for shifted Laplacians on the circle
Consider on the circle $S^1$ the operator
$$L := - \frac{\partial^2}{\partial \theta^2} + c$$
for some constant $c \in \mathbb{R}$.
What is its $\zeta$-regularized determinant?
This should be well-...
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Proof that derivative of Hurwitz Zeta by the first argument is not expressable in terms of Hurwitz Zeta
The set of elementary functions is defined so that it to be closed against operation of differentiation. It is also evidently close against discrete differentiation.
In the discrete calculus there is ...
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Bound of Chebyshev function and zeros of zeta function
It is an elementary argument (such as in Multiplicative Number Theory, section 18) that, if the Chebyshev's function $f(x) = \sum_{n \le x} \Lambda(x) = x + O(x^\alpha)$ for some $\alpha < 1$, then ...
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Analogues of the Riemann-Roch Theorem
In his thesis, Tate derives a Poisson formula on the adeles and a theorem which he calls the "Riemann-Roch Theorem". More specifically, given a continuous, $L^1$ function $f$ on the adeles such that ...
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A (likely) positivity property of the Lerch zeta-function
The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where
$$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$
is the Lerch zeta-...
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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, ...
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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 $\Re(s)>\alpha$ ...
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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 ...
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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:...
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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.