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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2-adic valuation of $L(0,\chi)$ for a Dirichlet character
Let $\chi : (\mathbb Z/f\mathbb Z)^\times \to K = \mathbb Q(\mu_{\phi(f)})$ be a primitive Dirichlet character. Assume moreover that it is not quadratic, that is, $\chi^2$ is not the trivial character....
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Asymptotic for number of prime elements in a number field
Let $K$ be a number field and let $O_K$ its ring of integers. Identify $K$ with its image in $\mathbb{C}^{\text{Hom}_{\mathbb{Q}-\text{alg}}(K,\mathbb{C})}$, which we consider equipped with the $|| \...
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Evaluating $\sum_{k=1}^{\infty} \frac{k}{2^k} \frac1{e^{t/{2^k}} + 1} $
If I arrive to calculate the sum $\displaystyle \sum_{k=1}^{\infty} \frac k{2^k} \frac{1}{\large e^{\frac t{2^k}} + 1} $ I think I can give a close form to the values $\zeta(1+\frac {2ip\pi}{\ln2})$ ...
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Compensation by the residue of the zeta function
(Repost of a question from MSE, where it found no success)
Let $F$ be a global number field. Introduce a local quantity at every place
$$x_p = \frac{\zeta_p(1)}{\zeta_p(2)}$$
for instance. The ...
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What are positive divisors of degree 2 on Elliptic Curve $y^2=x^3-x-1$ over $\mathbb{F}_3$?
Let (E) be an elliptic curve $y^2=x^3-x-1$ over $\mathbb{F}_3$, $a_0=1$, $a_n$ is the number of positive divisor of degree $n\geq 1$. $a_1$ in this case is the number of points of E, i.e., $a_1=1$ ...
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Voronin universality of random analytic functions with nontrivial zeros on a line
Recently a certain random analytic function was defined by probabilists: in an appropriate sense the limit of characteristic polynomials of random unitary matrices. Associated functions for other ...
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How does $\zeta^{\mathfrak{m}}(2)$ and relate to $\zeta(2)$?
EDIT There appears to be a numerical zeta function $\zeta(2)$ as well as at least two different "motivic" zeta function realizations (Betti and de Rham) $\zeta^{\mathfrak{m}}(2)$. The "period map" of ...
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Values of Artin L-functions at negative integers
Let $F$ be a number field and $\chi$ a one dimensional Artin character. That is, it is a map $\chi: Gal(\overline F/F) \to \mathbb C^\times$ and let $L(s,\chi)$ be it's L-series.
What is known about ...
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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 ...
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Zeta function of Abelian variety over finite field
Let $A/\mathbf{F}_q$ be an Abelian variety of dimension $g$. Suppose one knows $|A(\mathbf{F}_{q^n})|$ for all $1 \leq n \leq g$. Does one know then $\zeta(A,s)$ (equivalently, $|A(\mathbf{F}_{q^n})|$ ...
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Properties of $\zeta(s)\zeta(2s)\zeta(3s)...$
Let's consider the Dirichlet series $f(s)=\sum_{n=1}^\infty a_n n^{-s}$, where $a_n$ is the number of non-isomorphic abelian groups of order $n$. Now $a_n$ is weakly multiplicative and $a_{p^k}=P(k)=$ ...
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Critical values of L-functions and weights of Eisenstein Series
I have been reading Serre's paper on p-adic modular forms and there seems to be a connection between critical values of L-functions and weights of Eisenstein series in the following sense:
For the ...
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Interchanging limit and infinite product in Euler product for Dedekind function s=1
For an quartic (non-Galois) CM-field $K$ I have factors $v_p$ and for every prime $p$ found the following relation
$$v_p={\frac {\prod_{\mathfrak{p}|p;\mathfrak{p}\subset\mathcal0_{K}}(1-N_{{K/{\...
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Isospectrality, Gassmann-Sunada triples, and tensor products
It is well known that Gassmann-Sunada group triples can be used to construct isospectral manifolds, arithmetically equivalent number fields, etc. (Recall that a Gassmann-Sunada triple $(U,V,W)$ ...
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Analytic extension of the Hurwitz ζ function
For the purpose of formalisation in a theorem prover, I am looking for a simple definition of the analytic extension of the Hurwitz ζ function $\zeta(s,q)$ valid for all $s\in\mathbb{C}\setminus\{1\}$ ...
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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.
...
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$\zeta(0)$ of lichnerowicz operator on sphere quotients
Consider the lichnerowicz operator acting on symmetric traceless tensors on an even sphere $S^d$ with conical defect parametrized by angular deficit $\alpha$. Is there a simple way to understand/...
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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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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 ...
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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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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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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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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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Regularisation of $\sum_{n=0}^\infty \frac{1}{(a+n^2)^p}$
As from the title, I am currently dealing with this sum
$\sum_{n=0}^\infty \frac{1}{(a+n^2)^p}$
in particular with $p=1/2,3/2,...$ (but once solved for $p=1/2$ one can derive wrt $a$ and find the ...
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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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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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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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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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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 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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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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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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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
$$C_1=\sum_{1\...
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Zeros of the series $\phi(s, a) = \sum_{n \geq 0} e^{-an}(n + 1)^{-s}$
I have been recently interested in the series $\phi(s, a)$ of the title.
There, a is defined to be any positive real number and s is a complex variable.
The main reason for my curiosity is that the ...
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What does this connection between Chebyshev, Ramanujan, Ihara and Riemann mean?
It all started with Chris' answer saying returning paths on cubic graphs without backtracking can be expressed by the following recursion relation:
$$p_{r+1}(a) = ap_r(a)-2p_{r-1}(a)$$
$a$ is an ...
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Arithmetic zeta function and local zeta functions
For the arithmetic zeta function of (say) a nonsingular projective variety $X$, one has the following Euler product
\begin{equation}
\zeta_X(s) = \prod_{p\ \mbox{prime}}\zeta_{X\vert\mathbb{F}_p}(s),
...
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Local zetafunction of T-group and Lie-ring coincide
Grunewald, Segal and Smith define in "Subgroups of finite index in nilpotent groups" a zetafunction associated to a finitely generated, torsion-free nilpotent group G by counting normal subgroups ...
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What is the value of $p$-adic $\zeta$-function at positive integer point?
$p$-adic zeta function is a $p$-adic interpolation of the Riemann $\zeta$-function for the values $\zeta(1−k)$, $k\ge 1$ (see $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz) ...
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Computations of some character sums/zeta function
I'm currently trying to "compute" the zeta function of some hyperelliptic curves over a finite field (of odd characteristic). Precisely, let $b\in\mathbb{F}_q^\times$ be a non zero element (I also ...
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Asymptotics of A261668
In Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras, Proposition 10.8, Jianqiang Zhao mentiones the sequence:
$$a_n=\sum_{...