The zeta-functions tag has no wiki summary.

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### Why does the Riemann zeta function have non-trivial zeros?

This is a very basic question of course, and exposes my serious ignorance of analytic number theory, but what I am looking for is a good intuitive explanation rather than a formal proof (though a ...

**58**

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### 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 ...

**47**

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### Is there a “Basic Number Theory” for elliptic curves?

Tate's thesis showed how to profitably analyze $\zeta$ functions of number fields in terms of adelic points on the multiplicative group. In particular, combining Fourier analysis and topology, Tate ...

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### Are there refuted analogues of the Riemann hypothesis?

The classical Riemann Hypothesis has famous analogues for function fields and finite fields which have been proved. It has by now very many analogues, many of them still open. Are there important ...

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### Why does the Gamma-function complete the Riemann Zeta function?

Defining $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ yields $\xi(s) = \xi(1 - s)$ (where $\zeta$ is the Riemann Zeta function).
Is there any conceptual explanation - or ...

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### From Zeta Functions to Curves

Let $C$ be a nonsingular projective curve of genus $g \geq 0$ over a finite field $\mathbb{F}_q$ with $q$ elements. From this curve, we define the zeta function
$$Z_{C/{\mathbb{F}}_q}(u) = ...

**28**

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### 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 ...

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### 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 ...

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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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### A puzzling remark of Manin (ICM 1978)

Manin ends his 1978 ICM talk with this remark:
I would also like to mention I. M. Gel'fand's suggestion that the $\zeta$-functions of certain special differential operators should have an arithmetic ...

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### 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 ...

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### The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.

I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt ...

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### Universality of zeta- and L-functions

Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...

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### The class number formula, the BSD conjecture, and the Kronecker limit formula

If K is a number field then the Dedekind zeta function Zeta_K(s) can be written as a sum over ideal classes A of Zeta_K(s, A) = sum over ideals I in A of 1/N(I)^s. The class number formula follows ...

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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 "expectional zeros" of course first ...

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### What is the difference between a zeta function and an L-function?

I've been learning about Dedekind zeta functions and some basic L-functions in my introductory algebraic number theory class, and I've been wondering why some functions are called L-functions and ...

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### How was the importance of the zeta function discovered?

This question is similar to Why do zeta functions contain so much information? , but is distinct. If the answers to that question answer this one also, I don't understand why.
The question is this: ...

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### 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 ...

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### How does one motivate the analytic continuation of the Riemann zeta function?

I saw the functional equation and its proof for the Riemann zeta function many times, but ususally the books start with, e.g. tricky change of variable of Gamma function or other seemly unmotivated ...

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### Refinements of the Riemann hypothesis

I have often read that the Riemann hypothesis is somewhat a statement like:
The primes are as regularly distributed as we can hope for.
For example $\pi(x) = Li(x)+ O(x^{\sigma+\epsilon})$ for ...

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### 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 ...

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### Evaluating the integral $\int_{1}^{\infty}\frac{\{u\}}{u^{2}}\left(\log u\right)^{k}du.$

I am trying to find a formula for the following integral for non-negative integer $k$:
$$\int_1^{\infty}\frac{\{u\}}{u^{2}}\left(\log u\right)^{k}du.$$
My first thought was to use the formula ...

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### 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 ...

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### Can the failure of the multiplicativity of Euler factors at bad primes be corrected?

Warning: This one of those does-anyone-know-how-to-fix-this-vague-problem questions, and not an actual mathematics question at all.
If $X$ is a scheme of finite type over a finite field, then the ...

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### Dirichlet's regulator vs Beilinson's regulator

Consider a number field $F$ with ring of integers $O_F$. The Beilinson regulator can in this particular setting be viewed as a map from $K_n(O_F)$ to a suitable real vector space. Here $n$ is any ...

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### Is there an elegant algebraic proof of this formula for quadratic field discriminants?

Consider the Dirichlet series counting discriminants of real quadratic fields. Quadratic field discriminants are "basically" squarefree integers, so the associated Dirichlet series $\sum D^{-s}$ is ...

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### Dwork's use of p-adic analysis in algebraic geometry

Using p-adic analysis, Dwork was the first to prove the rationality of the zeta function of a variety over a finite field. From what I have seen, in algebraic geometry, this method is not used much ...

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### Does this sum equal zeta(3)?

Does:
$$\sum_{1 \leq i<j} \frac{1}{i j^2} = \sum_{1 \leq k} \frac{1}{k^3}?$$
Motivation: Call the above sum $S$, and let
$$T := \sum_{ GCD(i,j)=1} \frac{1}{\max(i,j) i j}.$$
The sum $T$ came up ...

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### 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 ...

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### PNT for general zeta functions, Applications of.

When I read it for the first time, I found the whole slog towards proving the Prime Number Theorem and the final success to be magnificent. So I am curious about more general results.
We talk of ...

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### Is there an analogue of the Lefschetz fixed point theorem for discrete dynamical systems?

Background/Motivation
Let $(X, f)$ be a discrete dynamical system. For now, $X$ is just a set and $f$ is just a function $f : X \to X$. Suppose that $f^n$ has a finite number of fixed points for ...

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### non-trivial zeros of partial zeta functions

Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as
$$
\zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s}
$$
where $Re(s)>1$. Let $\omega$ be either ...

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### Lower bounds on zeta(s+it) for fixed s

This is most probably widely known and discussed here many times, so I am preliminay sorry.
Does Riemann conjecture imply some lower estimates on values, say $|\zeta(3/4+it)|$ for real $t$, when ...

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### Sets with zeta functions that are not the primes

Does there exist a set $S \subset \mathbb N$ such that the Dirchlet density of $S$ is well-defined and positive, the Dirchlet density of $S \cap \operatorname{PRIMES}$ is well-defined and zero, and:
...

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### Behaviour of Zeta-function under Finite Morphism

Let X ---> Y be a finite surjective morphism of smooth, projective, connected varieties over a finite field F_q. Can one describe the zeta function Z(X, t) in terms of the zeta-function Z(Y,t) of ...

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### Establishing zeta(3) as a definite integral and its computation.

I am a 19 yr old student new to all these ideas. I made the transformation $X(z)=\sum_{n=1}^\infty z^n/n^2$. Therefore $X(1)=\pi^2/6$ as we all know (it is $\zeta(2)$). To calculate $X(1)$, I ...

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### Weil Conjectures for nonprojective algebraic varieties

If we replace projective variety with algebraic variety in the statement of the Weil conjectures what happens? To me it seems the statement still makes sense. But is it still true?

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### Periods and L-values

A famous theorem of Euler is that Zeta(2n) is a rational number times pi^(2n). Work of Kummer, Herbrand, Ribet and others shows that the rational multiplier has number theoretic significance.
For ...

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### Degree of Transcendentality and Feynman Diagrams

Physicists computing multiloop Feynman diagrams have introduced various
techniques and conjectures that involve the notion of Degree of Transcendentality (DoT). From what I understand one defines
1) ...

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

Someone recently quoted to me this recent article that claims to prove that $\zeta(2n+1) \notin (2\pi )^{2n+1} \mathbb{Q}$.
I always assumed this was well known. More precisely I thought this result ...

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### L-Functions of Varieties, Zeta Functions of Their Models

Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the ...

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### Is it easy to prove that $\sum_n |X(\mathbb{F}_{q^n})| t^n$ is rational?

Background: Let $X$ be an algebraic variety over a finite field $\mathbb{F}_q$. One of the successes of Etale cohomology - previously achieved by Dwork- was proving the rationality of the Zeta ...

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### A question on K_1 of an elliptic curve

Consider an elliptic curve $E/ \mathbb{Q}$, with a regular model $\mathcal{E} / \mathbb{Z}$. We have (Beilinson regulator) maps
$$ K_1(\mathcal{E})^{(2)} \to K_1(E)^{(2)} \to H_D^3(E_{/ \mathbb{R}} , ...

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### Wick rotation and the Riemann zeta function

The goal of this question is to conceptualize in some way the fact that the Riemann zeta function $\zeta(s)$, and other zeta functions like it, have analytic continuations.
Background
I have by now ...

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### Euler product over primes congruent to 3 mod 4

I'm a string theorist and I have come across the following expression in a computation I'm doing (involving a sum over inequivalent Lens spaces):
$$\widehat{\zeta}(s)=\prod_{\mathrm{primes}\ p\equiv ...

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### 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 ...

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### How does the order of a pole of a zeta function indicate any geometric information?

Here, I'm primarily concerced about zeta functions of hypersurfaces over fields of finite characteristic.
Assume $F_q$ to be a finite field with q elements. Consider the zeta function of the ...

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### is there a p-adic Borel theorem?

Let $F$ be a number field. Denote, as usual, $\mathcal{O}_F$ the ring of integers and $r_1$, $r_2$ the number of real and complex embeddings. Let $\zeta_F(s)$ be the Dedekind zeta function of $F$. ...

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### Is it always possible to compute the Betti numbers of a nice space with a well-chosen Lefschetz zeta function?

Let $X$ be a smooth projective variety. If I've understood correctly, the Weil conjectures imply that it is possible to compute the Betti numbers of $X(\mathbb{C})$ by computing the local zeta ...

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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, ...