# 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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### 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}...

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### 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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### 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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### 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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### 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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### $\zeta(2n+1)$ - 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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### Computing the eta invariant of a rather contrived operator on the circle

For physical reasons, I am interested in computing the eta invariant of the following Hermitian operator acting on complex valued functions on the circle with circumference 1. I define the operator ...

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### When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ with $\Re(\zeta(s))\neq 0$ and $\Im(\zeta(s))\neq 0$? [closed]

When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ for $0<\Re(s)<1$. Here $\zeta$ denotes the Reimann zeta function. Does the solution live on a vertical line? Or is this another coincidence when both ...

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### 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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### 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 ...

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### 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 ...

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### 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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### 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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### 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\...

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### 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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### 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 ...

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### Derivative of zeta at positive even integers

Is there a general formula that sums up all values of $ζ′(2n)$, such that $n\in\mathbb{N}$?

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### 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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### 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 ...

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### $L$-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\...

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### 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 $...

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### 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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### 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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### 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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### Explicit formula for zeta function with special type of weight

Consider the following line of thinking:
$$\pi(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x} $$
Here,
$\operatorname{R}(x) = \...

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### Have Bushnell and Reiner got it wrong?

Let $M$ be a finite-dimensional simple $\mathbb Q$ algebra and $\Lambda$ an order in $M$. Its zeta-function is defined as
$$
Z(s)=\sum_{I}|\Lambda/I|^{-s},
$$
where the sum runs over all left ideals ...

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### Eisenstein series evaluated at $2i$

Consider the real analytic Eisenstein series defined by
$$
E(z,s) := \sum_{\gamma\in\Gamma_\infty\setminus\Gamma} Im(\gamma z)^s
$$
where as usual, $\Gamma=SL(2,\mathbb{Z})$ and $\Gamma_\infty$ is the ...

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### Are the zeroes of the finite characteristic zeta functions dense in $\left\{s\in\mathbb{C}\mid\mathfrak{Re}(s)=\frac{1}{2}\right\}$?

If $p$ is a prime, $n\in\mathbb{N}$ is a natural number and $C$ is a nonsingular curve over $\mathbb{F}_{p^{n}}$, the $\zeta$ function associated to $C\mid_{\mathbb{F}_{p^{n}}}$ is defined as
\begin{...

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### Class of spectral zeta functions whose analytic extension takes a particular form

In quantum field theory the one-loop effective action is expressed in terms of the functional determinant of the (elliptic and self-adjoint) operator of small disturbances. Since the real eigenvalues ...

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### Growth of residues of $1/\zeta(s)$: conjectures?

Let $\rho$ range over the non-trivial zeroes of the Riemann zeta function. Let
$$M(T) = \max_{|\Im \rho|\leq T} \left|\mathrm{Res}_{s=\rho} \frac{1}{\zeta(s)}\right| =
\max_{|\Im \rho|\leq T} \frac{1}...

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### Connecting two pictures of the Zeta function

Lets consider two views of zeta functions of curves.
For the following, let $\mathbb{F}_p$ be the field with $p$ elements where $p$ is prime, and let $\overline{\mathbb{F}_p}$ be the algebraic closure ...

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### Zeta functions of schemes of finite type over $\mathbb{Z}$

Let $X$ be a scheme of finite type over $\mathbb{Z}$. In Section 11 of my 2008 paper in J. Number Theory, "Ring structures on groups of arithmetic functions," I define an additive analogue $...

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### Constant coefficient of Eisenstein series

Let $\chi$ be a character of $\mathbb{Q}^{\times}\setminus\mathbb{A}^{\times}$. We define an induced representation of $Mp_2(\mathbb{A})\simeq SL_2(\mathbb{A})\times \mathbb{C}^1$,
$$I(s,\chi) := \{\...

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### Consequences of infinitely many double zeros of zeta function of number field

Related to this and this.
Let $K$ be the number field with the degree 24 defining polynomial
...

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### Double zero at $1/2$ of zeta function of number field

Ten years old question asks about Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field.
Let $K$ be the number field with the degree 24 defining polynomial
...

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### How essential is the vanishing of the Dirichlet $L$-functions to Dirichlet's theorem on primes in arithmetic progressions?

I seem to recall that the prime number theorem (PNT) is equivalent to the fact that the Riemann zeta function $\zeta(s)$ is non-zero on all of $\text{Re}(s) = 1$ (see https://math.stackexchange.com/...

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### Galois group of zeta function of hyperelliptic curve

Let $f \in \mathbb F_q[T]$ be monic, squarefree.
Can we say anything on the Galois group of $Z_f$, the zeta function of the hyperelliptic curve $y^2=f$, directly in terms of $f$ (coefficients or ...

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### Selberg zeta function analytic expressions

Consider the following algebraic equation,
$$
y^n=\frac{(z-z_1)(z-z_3)}{(z-z_2)(z-z_4)}
$$
which is a Riemann surface of genus $n-1$ (after compactifying). The classical retrosection theorem due to ...

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### Class groups and zeta functions for maximal orders in CSAs

I'm looking for references for certain algebraic objects in the context of maximal orders in (finite dimensional) central simple algebras over algebraic number fields. Does anyone know of any good ...

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### Asymptotically similar functions with opposite parity, were they considered, are they useful? Case of polynomials

So, can we transform an even function into an odd function and vice versa?
Let's consider this method:
Transformation even->odd:
Suppose $f_{even}(x)$ is a function which satisfies the following ...

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### Moduli space of germs of riemannian metrics

Let $S$ be the set of germs of riemannian metrics near $0$ on $\mathbb R^n$. It is acted on by the group $\textrm{Diff}$ of germs of diffeomorphisms of $\mathbb R^n$ preserving $0$.
Let's denote by $S^...

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### Schemes with common zeta function

If $S_\zeta$ is the set of all separated schemes of finite type over $\mathbb{Z}$ that have the same arithmetic zeta function $\zeta$, what more can we say about $S_\zeta$ assuming it is non-empty?

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### Update on "A Mad day's work" by Cartier

In his paper "A mad day's work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry," Cartier discusses in the last sections 8 and 9 the role of ...

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### Riemann-Siegel formula for Dirichlet characters

After unearthing and giving a proof of what is now known as the Riemann--Siegel formula for the Riemann zeta function enabling the computation of $\zeta(1/2+iT)$ in time $O(T^{1/2})$,
in 1943 Siegel ...

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### On integral relating logarithm of absolute value of Zeta function

Sorry for such a direct question:
Consider the following integral:
$$I(t)=\int_{1/2}^{1} {\log|\zeta(a+it)|}da.$$
How to find the nature of $I(t)$ as $t\rightarrow\infty$?

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### Dedekind Zeta functions of Biquadratic fields

Let $F/ \mathbb{Q}$ be a biquadratic field of Galois group $C_2 \times C_2$. Then I know that the Dedekind Zeta function of $F$ can be factored into $L$-functions as; $$\zeta_F(s) = \zeta(s) L(s, \...

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### Why is the regulator of the degree 11 extension of $\mathbb{Q}$ so large?

Let $K_\infty/\mathbb{Q}$ denote the $\hat{\mathbb{Z}}$-extension of $\mathbb{Q}$. Then for each $n\geq1$, $K_\infty$ has a unique subfield $K_n$ of degree $n$ over $\mathbb{Q}$.
The fields $K_n$ are ...

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### Contemporary introduction to Godement-Jacquet "Zeta functions of simple algebras"

The question is in the title: The book
Godement-Jacquet "Zeta functions of simple algebras"
is from 1971. Has there ever been a textbook introduction to this material, or at least part of it?...

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### Relation between Schanuel's theorem and class number equation

(Crossposted on math stack exchange: https://math.stackexchange.com/questions/4040249/relation-between-schanuels-theorem-and-class-number-equation)
It was recently brought to my attention that there ...

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### Serge Lang's proof of Brauer-Siegel theorem

I was reading through chapter 16 of Lang's Algebraic Number Theory book. The chapter is fully devoted to proving the Brauer-Siegel theorem: Let ${(k_n/ \mathbb{Q})}_n$ be a sequence of galois ...