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.
325 questions
1 vote
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Čech cohomology satisfies descent if the underlying spectrum universally bounded from below
Let $X\longrightarrow\operatorname{Spec}(k)$ be an étale scheme of finite relative dimension $d$, $U\subseteq X$ an affine open subscheme, $\mathcal{U}$ a hypercovering consisting of affine schemes. ...
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1 vote
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Behavior of the spectral sequence associated to topological Hochschild homology
For any presheaf $F$ on a site $X$ let $\eta_{F}$ be the composite map \begin{equation*} \eta_{F}(U):F(U)\longrightarrow\prod_{u\in U}F_{u}\longrightarrow H^{0}(U,F)\longrightarrow H^{\bullet}(U,F)...
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2 votes
1 answer
205 views
Isomorphisms of sheaves of Abelian groups in Hesselholt's "Topological Hochschild Homology and the Hasse-Weil Zeta function"
This is from §5, p. 13, of Hesselholt's "Topological Hochschild Homology and the Hasse-Weil Zeta function". The author claims there is a family of isomorphisms \begin{equation*} \phantom{\...
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3 votes
0 answers
125 views
Is there an algebraic notion of two primes being close?
I apologise in advance for the vagueness of the question below. I am not at all an expert in algebraic geometry, it might be that the question will come across as very naive, sorry! I was wondering ...
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2 votes
0 answers
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Distribution of lengths of closed geodesics
(I am trying to get a sense of what the state of the art is regarding the distribution of the length spectrum of a closed surface of negative curvature, I am curious about any good reference/open ...
- 3,020
2 votes
0 answers
81 views
Dynamical systems of twisted Ihara zeta functions on graphs
A twisted Ihara zeta function of a finite connected graph $\Gamma$ is sensitive to a quantity known as holonomy. The zeta function can be defined as: $$\zeta_{\Gamma}(u, \rho) := \prod_{[P]} \left(1 - ...
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1 vote
0 answers
102 views
Dirichlet series from second differences of recursively summed prime gaps
I'm investigating a Dirichlet series built from a recursively summed and differenced sequence of prime gaps. $\text{Let } g_n = p_{n+1}-p_n$, denote the prime gaps. From these, construct: $$S_0(n)=g_n,...
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6 votes
0 answers
187 views
Shared large partial quotient and modular-like relations in continued fractions of ζ(3)/(m² log π)
Update: Two additional unusual continued fraction behaviors have been observed for expressions involving zeta(3) and log(pi) and are documented in the second script below. I've recently encountered ...
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0 votes
0 answers
197 views
Why are there no L-factors corresponding to the infinte places for $\zeta$ functions over $\mathbb{F}_{p^{n}}$?
Deninger has constructed $L$-factors $\zeta_\infty(s) := 2^{-1/2} \pi^{-s/2} \Gamma(s/2)$, that are to be interpreted as "$L$-factors at infinity". The conjectural "Deninger Trace ...
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Graphs and reflection-induced color symmetries
Let $\Gamma = (V, E)$ be a finite, planar labeled, undirected, tailless, topological multigraph ($4$-regular) arising from a $z=0$ slice of a stratified foliated manifold. The graph admits a ...
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3 votes
1 answer
248 views
Understanding choice of smoothing function in Titchmarsh
I am currently reading Chapter 4 of Titchmarsh's Theory of the Riemann zeta function, about Approximate Formulae. The following theorem is given. Theorem: We have \begin{equation*} \zeta(s)^k=...
user553077
3 votes
0 answers
118 views
Is there a "multiple" Igusa zeta function?
Let $[K:\mathbb Q_p]<\infty, R=\mathcal{O}_K$, and $\pi$ a uniformizer. The Igusa zeta function is defined for a Schwartz--Bruhat function $\phi:K^n\to \mathbb C$, a character $\chi:R^\times\to \...
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3 votes
0 answers
101 views
Relation between left ideal and right ideal zeta functions of orders
We try to keep things as simple as possible. Let $A$ be a $\mathbf{Q}$-algebra which is a simple ring of dimension $n$ over $\mathbf{Q}$. Let $\mathfrak{O}\subseteq A$ be an order, i.e. a subring ...
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0 votes
0 answers
104 views
The Hurwitz zeta function on the line $\mathrm{Re}(s) = 1$
For $\mathrm{Re}(s) > 1$ and $0 < a \leqslant 1$, let $\zeta(s,a) = \sum_{n \geqslant 0} 1/(n+a)^s$ denote the Hurwitz zeta function. As a function of $s$, $\zeta(s,a)$ has a meromorphic ...
2 votes
1 answer
153 views
Equivalence of an iterated integral and a multiple zeta value
As I understand, the multiple zeta function is related to Chen's iterated integral in the following way: $$ \zeta(s_1, s_2, \dots, s_k) = \int_{0}^1 \frac{1}{t_1^{s_1 - 1}} \int_{0}^{t_1} \frac{1}{1 - ...
2 votes
0 answers
125 views
Zeta Function Regularization of a Bessel-Related Spectrum, WKB Approximation
I am reading through Steiner's 1987 paper "Spectral Sum Rules for the Circular Aharonov-Bohm Quantum Billiard" link. I am interested in section 5 (specifically 5.3) and I can derive 5.21 and ...
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0 votes
0 answers
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Is there a relation of topological entropy of a map "T" to Artin-Mazur zeta function of "T" ? (Same for Weil,Riemann,Ruelle,... zetas ?)
Given a map "T" topological space to itself (say compact) one may consider topological entropy of "T" (it is related to Kolmogorov–Sinai, or metric entropy in case of metric spaces)...
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0 votes
2 answers
232 views
Is $1+\left (\frac{m}{m-1}\right )^{\frac{\log(p-1)}{\log(2)}}\cdot(p-1)^{\frac{\log(m)}{\log(2)}+1}<\frac{m}{m-1}p^{\frac{\log(m)}{\log(2)}+1}$?
For $p>2,m>2$, is $$1+\left (\frac{m}{m-1}\right )^{\frac{\log(p-1)}{\log(2)}}\cdot(p-1)^{\frac{\log(m)}{\log(2)}+1}<\frac{m}{m-1}p^{\frac{\log(m)}{\log(2)}+1}$$ ? Motivation: I am trying to ...
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4 votes
0 answers
142 views
Euler factors from bad primes and the Beilinson-Bloch vanishing conjecture
The vanishing part of the Beilinson-Bloch conjecture asserts that for a smooth projective variety $X$ over a number field $K$, $\dim_{\mathbb{Q}} \operatorname{CH}^i(X) \otimes_{\mathbb{Z}} \mathbb{Q} ...
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0 votes
0 answers
205 views
Show that the function $(n+2)\zeta(n+3)-\zeta(n+2)-n-1$ is positive on $\mathbb{N}.$
I'm trying to prove the positivity of the function $(n+2)\zeta(n+3)-\zeta(n+2)-n-1$ on $\mathbb{N}$ using the inequalities \begin{equation*} \frac{s+1}{s}\zeta(s)\zeta(s+2)\geq \zeta^2(s+1),\quad s>...
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2 votes
1 answer
224 views
Is the function $(z-1)(2^{z}-1)\zeta(z)$ logarithmically concave and convex in $z\in(0,\infty)$?
For proving that the sequence \begin{equation}\label{first-proof-decreas-seq} \frac{1}{(2k-1)(k+1)} \frac{2^{2k+2}-1}{2^{2k}-1} \biggl|\frac{B_{2k+2}}{B_{2k}}\biggr| \end{equation} is decreasing in $k\...
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57 votes
2 answers
3k views
Is it known? A sum over lattice parallelograms of area one is equal to $\pi$
I recently discovered a formula, my proof is really a high school proof in three lines. $$4\sum_{x, \, y \, \in \, \mathbb Z_{\geq 0}^2, \, \det(x \ \ y) = 1} \frac{1}{\lVert x\rVert^2\cdot\lVert y\...
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6 votes
0 answers
384 views
Deeper meaning behind the occurrence of the factor $\frac{\log q}{i}$ in Deninger's results
In two papers Deninger proved the following: If $q=p^{n}$ and $p$ is a finite prime of $\mathbb{Z}$, $B=\mathbb{C}[\mathbb{C}]$ is generated by symbols of the form $e^{\alpha}$, $\alpha\in\mathbb{C}$,...
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1 vote
0 answers
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Particular values of multiple zeta functions of even weight
I'm new to the fascinating field of multiple zeta values. I just want to know if there are known evaluations for $\zeta(6,2)$ or $\zeta(9,3)$ or other particular case when the weight of the multiple ...
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4 votes
0 answers
414 views
A question on vanishing of Godement–Jacquet-like zeta integral
$\DeclareMathOperator\GL{GL}$The classical Godement–Jacquet zeta integral is of this form: $f$ is a matrix coefficient of a cuspidal automorphic representation of $\GL_n(\mathbb{A}_\mathbb{Q})$, and $\...
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12 votes
1 answer
939 views
$\zeta(-n)=2^{r_1}\frac{|K_{2n}(O)|}{|K_{2n+1}(O)|} R_K$ and replace $K$-theory with $\mathbb{S}$
There seems to be an agreement among experts that the formulas by Lichtenbaum in the 70's $$\zeta(-n)=2^{r_1}\frac{|K_{2n}(O_K)|}{|K_{2n+1}(O_K)|} R_K$$ follow from the resolution of the Bloch-Kato ...
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1 vote
0 answers
329 views
A question about decomposition of irreducible root system
Fix an irreducible root system $\Phi$ with rank $r$ and a root base $\Delta$ (we only care type ADE). Call a disjoint union $\Phi = \Phi_{1}\sqcup\dotsb\sqcup\Phi_{s}$ a decomposition of $\Phi$ if ...
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1 vote
0 answers
145 views
Where have you encountered the following arithmetic function?
The following arithmetic function is studied by Zagier in connection with values at odd negative integers of zeta functions of real quadratic fields: $$e_r (n)=\sum_{\underset{|x|\leq n}{x^{2}\equiv n\...
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-6 votes
1 answer
202 views
Is the real part of the Eta function bounded by $2 \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}} $ [closed]
Consider the series defined by \begin{equation} f(\alpha,\beta) := \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}\cos(\beta\ln(n))} \end{equation} is it true that $$f(\alpha,\beta) \le 2\sum_{n=1}...
24 votes
1 answer
2k views
What are $L$-functions?
I am coming at this question from the point of view of someone who is working in arithmetic geometry around the Langlands program. We have $L$-functions associated to many different structures that we ...
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2 votes
0 answers
171 views
Which ζ-function controls the density of base=2 Miller–Rabin pseudoprimes?
Background: the density d (≈ inverse distance between elements of the set) of primes is “controlled by zeros of ζ-function” in the sense that $(d(x) - T(x))·\log(x)/√x$ is an almost periodic function ...
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7 votes
0 answers
214 views
Special value of the Artin--Mazur zeta function in arithmetic dynamics
Let $X$ be a compact manifold and $f: X\rightarrow X$ be a diffeomorphism. Assume that the $k$-fold iterate $f^k: X\rightarrow X$ has finitely many fixed points for all natural numbers $k$. The ...
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2 votes
0 answers
158 views
Arithmetic interest of the Goss zeta function
I'm someone with more of a number fields background who recently started working on a project more in the function fields setting. I was reading Goss's book (Basic structures of function field ...
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3 votes
1 answer
660 views
$\zeta(2n+1)$ - Is this formulation helpful?
Cross-posting alert: I posted This on MSE. I read through the guidelines for cross-posting on both the sites and my conclusion is that I am not violating any guidelines. Briefly, I derived some ...
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7 votes
0 answers
350 views
Simple/Elementary derivation of Ramanujan's continued fraction for Hurwitz $\zeta(3,x+1)$
I came across this MSE post discussing a certain continued fraction for $\zeta(3)$ (more specifically, the Hurwitz zeta function $\zeta(s,z)$ at $s=3$) due to Ramanujan. I asked the original poster ...
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0 votes
1 answer
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Expanding on a step in the calculation of ζ(f,χ,s) = Λ(s)
I'm trying to understand the derivation here: Above, f is defined as a product of Schwartz-Bruhat functions which are their own Fourier transform. I was hoping someone could spell out the second ...
user30211
0 votes
0 answers
130 views
Relating the multiplicative Fourier transform and the derived characteristic polynomial
(Tuesday, Sept 5:) For a number field $Fˣ$ and a number ring $Oˣ$ it is common to define: $Z(f,χ) = ʃ_{Fˣ} f(x) χ(x) dˣ x$ $g(ω,ψ) = ʃ_{Oˣ} ω(x) ψ(x) dˣ x$ where $dˣx$ is the multiplicative Haar ...
user30211
2 votes
0 answers
245 views
Zeta function associated with a function $f$
Let the function $f(t) = \cos(at)$, where ($0 < a < 1$). Let us define $$\zeta(z, f) = \frac{1}{\Gamma(z)} \int_0^{+\infty} \frac{t^{z-1}\cos(at)}{e^t-1}\, dt. $$ Is there a general formula that ...
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5 votes
0 answers
213 views
Epstein zeta function for non-fundamental discriminant to L-series
Let $Q(x,y) = ax^2+b xy + cy^2$ be a primitive integral positive-definite quadratic form, with associated number field $K$. If $D=b^2-4ac$ is a fundamental discriminant, then it's well-known that $$\...
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0 votes
1 answer
239 views
Residue calculation for Eulerian expansion of the cotangent
I am looking for ideas on proving the Eulerian expansion of the cotangent using residue calculation: $$\pi\cot(\pi z)=\frac{1}{z}+\sum_{n=1}^{\infty}\left(\frac{1}{z+n}+\frac{1}{z-n}\right), \ z\in\...
- 461
7 votes
2 answers
1k views
Positivity of the coefficients of Taylor series associated to the Riemann hypothesis
The question below relates to the paper "Jensen Polynomials for the Riemann Zeta Function and Other Sequences" of Griffin, Ono, Rolen and Zagier. I'm asking it here because I am sure the ...
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3 votes
1 answer
1k views
Derivative of the Riemann zeta function at $z=-2$
I have a question regarding the derivatives of the Riemann zeta function. It is known that $\zeta'(-1)=\frac{1}{12}-\ln A$, where $A$ is the Glaisher-Kinkelin constant (which is an elegant ...
- 461
3 votes
1 answer
361 views
Derivative of zeta at positive even integers
Is there a general formula that sums up all values of $ζ′(2n)$, such that $n\in\mathbb{N}$?
- 461
2 votes
0 answers
95 views
Reference request for literature on the following function--power counting zeta function
I'll start by writing the character of interest, and describing some properties of it, before I get to the meat of the question. Any help is greatly appreciated, even an offhand suggestion/comment/...
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0 votes
1 answer
297 views
Can I calculate congruent zeta function of given hyperelliptic curve by hand?
How can I calculate the numerator of congruent zeta function of given hyperelliptic curve ? For example, let $C:y^2=(x^2+1)(x^4-8x^3+2x^2+8x+1)$. numerator of congruent zeta function mod$23$ of this ...
- 1,524
2 votes
0 answers
178 views
$L$-series and Riemann zeta function
I am currently reading SGA 4$\frac{1}{2}$, exposé 2: Rapport sur la formule des traces. The $L$-series associated to a scheme $X$ of finite type over $\mathbb{F}_{p}$ is defined as $$L(X,s):=\prod_{x\...
- 1,889
0 votes
0 answers
164 views
Is $p^{-s}$ transcendental if $\zeta(s)=0$?
Let $K$ be a number field and $\mathcal{O}_{K}$ its ring of integers. Let $$\zeta_{K}(s)=\prod_{\mathfrak{m}}\frac{1}{1-\#(\mathcal{O}_{K}/\mathfrak{m})^{-s}}$$ be the $\zeta$ function associated to $...
- 1,889
4 votes
1 answer
584 views
Proving $\zeta_K\left(\frac12\right)\neq 0 \implies \zeta_K'\left(\frac12\right)\neq0?$
For an algebraic number field $K$, let $\zeta_K(s)$ be the Dedekind zeta function associated to $K$, and let $\zeta_K'(s)$ be its derivative. I believe that the following statement is true: $$\zeta_K\...
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27 votes
4 answers
4k views
Why do we care about the eigenvalues of the Frobenius map?
The Riemann hypothesis for finite fields can be stated as follows: take a smooth projective variety X of finite type over the finite field $\mathbb{F}_q$ for some $q=p^n$. Then the eigenvalues $\...
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4 votes
0 answers
178 views
Converse theorem for zeta universality
Voronin's Universality Theorem for $\zeta(s)$ is that the zeta function can uniformly approximate any non-vanishing holomorphic function to any degree of accuracy in the right-half of the critical ...
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