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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3 answers
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Non-vanishing of zeta(s), Re(s)=1, without complex analysis?

Say you are allowed to use Fourier analysis, complex variables, Euler-Maclaurin, etc., but no complex analysis - no holomorphic continuations, no definition of analytic function, and, in particular, ...
0 votes
0 answers
54 views

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 ...
0 votes
2 answers
477 views

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 $t\rightarrow\infty$?
7 votes
0 answers
242 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 ...
-3 votes
2 answers
299 views

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 ...
3 votes
1 answer
392 views

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 ...
0 votes
1 answer
105 views

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 ...
0 votes
0 answers
86 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 ...
106 votes
7 answers
20k views

What is the field with one element?

I've heard of this many times, but I don't know anything about it. What I do know is that it is supposed to solve the problem of the fact that the final object in the category of schemes is one-...
1 vote
0 answers
458 views

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) = \...
2 votes
0 answers
212 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 ...
7 votes
2 answers
822 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 ...
4 votes
0 answers
139 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 $$\...
0 votes
1 answer
144 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\...
3 votes
1 answer
537 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 ...
19 votes
2 answers
2k views

Dedekind Zeta function: behaviour at 1

Let $\zeta_F$ denote the Dedekind zeta function of a number field $F$. We have $\zeta_F(s) = \frac{\lambda_{-1}}{s-1} + \lambda_0 + \dots$ for $s-1$ small. Class number formula: We have $\lambda_{-...
3 votes
1 answer
287 views

Derivative of zeta at positive even integers

Is there a general formula that sums up all values of $ζ′(2n)$, such that $n\in\mathbb{N}$?
2 votes
0 answers
77 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/...
0 votes
1 answer
204 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 ...
2 votes
0 answers
148 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\...
38 votes
5 answers
7k views

Is $\zeta(3)/\pi^3$ rational?

Apery proved in 1976 that $\zeta(3)$ is irrational, and we know that for all integers $n$, $\zeta(2n)=\alpha \pi^{2n}$ for some $\alpha\in \mathbb{Q}$. Given these facts, it seems natural to ask ...
17 votes
1 answer
3k views

Values of zeta at odd positive integers and Borel's computations

Someone recently quoted to me this recent article that claims to prove that $\zeta(2n+1) \notin (2\pi )^{2n+1} \mathbb{Q}$. [Edit: published reference: Musha, Takaaki. Negation of the conjecture for ...
4 votes
1 answer
503 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\...
0 votes
0 answers
138 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 $...
26 votes
4 answers
3k 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 $\...
29 votes
4 answers
5k views

What is Ricardo Pérez-Marco's eñe product? Does it explain his statistical results on differences of zeta zeros?

The number theory community here at University of Michigan is abuzz with talk of this paper recently posted to the arxiv. If you haven't seen it already, the punch line is that the global differences ...
16 votes
8 answers
4k views

Brownian bridge interpreted as Brownian motion on the circle

Is it reasonable to view the Brownian bridge as a kind of Brownian motion indexed by points on the circle? The Brownian bridge has some strange connections with the Riemann zeta function (see Williams'...
39 votes
2 answers
2k views

Is there a "quantum" Riemann zeta function?

Occasionally I find myself in a situation where a naive, non-rigorous computation leads me to a divergent sum, like $\sum_{n=1}^\infty n$. In times like these, a standard approach is to guess the ...
4 votes
0 answers
124 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 ...
9 votes
5 answers
3k views

Negative values of Riemann zeta function on the critical line.

From parametric plots of $\zeta \left( \frac{1}{2} + it \right)$ it seems to be the case that: (1) except for $\zeta(\frac{1}{2})$ the Riemann zeta function does not attain any negative real value on ...
2 votes
0 answers
149 views

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 ...
0 votes
0 answers
93 views

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{...
1 vote
2 answers
825 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: https://mathoverflow.net/questions/115179/real-root-1-of-the-hasse-weil-l-function-of-c-over I am just curious to know ...
1 vote
0 answers
58 views

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 ...
4 votes
1 answer
237 views

Are there any extensive treatments on rational zeta series?

I've been trying to find an extensive, in-depth treatment of rational zeta series. Via the Wikipedia article on the topic, I've found two articles on this subject. While they are certainly very ...
14 votes
0 answers
789 views

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}...
2 votes
1 answer
332 views

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 ...
39 votes
10 answers
6k views

Why are functional equations important?

People who talk about things like modular forms and zeta functions put a lot of emphasis on the existence and form of functional equations, but I've never seen them used as anything other than a ...
1 vote
0 answers
176 views

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 $...
14 votes
0 answers
929 views

Witten zeta function v.s. Riemann zeta function

From a talk, we learned that The symplectic volume of the space $M$ of gauge equivalence classes of flat G connections is given by the “Witten zeta function”: where we sum over irreducible ...
1 vote
0 answers
218 views

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) := \{\...
0 votes
1 answer
202 views

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 ...
6 votes
0 answers
210 views

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 ...
19 votes
3 answers
2k views

Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field.

Let $M$ be the splitting field of x^8 + 3*x^7 + 13*x^6 + 17*x^5 + 45*x^4 + 37*x^3 + 11*x^2 + 112*x + 108 over the rationals. If I've understood some tables ...
2 votes
1 answer
415 views

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/...
6 votes
0 answers
168 views

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 ...
5 votes
0 answers
128 views

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 ...
1 vote
0 answers
57 views

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 ...
0 votes
0 answers
173 views

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 ...
11 votes
1 answer
794 views

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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