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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Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta ...
Tom Copeland's user avatar
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37 votes
8 answers
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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 usually the books start with, e.g. tricky change of variable of Gamma function or other seemingly unmotivated ...
36min's user avatar
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29 votes
4 answers
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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 "exceptional zeros" of course first ...
Kálmán Kőszegi's user avatar
72 votes
10 answers
17k views

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 ...
Peter Arndt's user avatar
41 votes
3 answers
4k views

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) = \exp\left(\...
Peter Humphries's user avatar
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-...
Benjamin Antieau's user avatar
34 votes
2 answers
3k views

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 ...
Kevin Buzzard's user avatar
104 votes
6 answers
19k views

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 ...
gowers's user avatar
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74 votes
4 answers
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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 ...
Timothy Chow's user avatar
56 votes
4 answers
5k views

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 ...
Colin McLarty's user avatar
49 votes
3 answers
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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 ...
David Hansen's user avatar
33 votes
1 answer
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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 ...
draks ...'s user avatar
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26 votes
3 answers
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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 ...
M.G.'s user avatar
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20 votes
4 answers
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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 ...
vamsi krishna's user avatar
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 ...
Kevin Buzzard's user avatar
16 votes
0 answers
847 views

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 ...
Tom163's user avatar
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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 ...
joro's user avatar
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63 votes
3 answers
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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 ...
David E Speyer's user avatar
29 votes
5 answers
6k views

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 ...
David Corwin's user avatar
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19 votes
3 answers
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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 ...
wood's user avatar
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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_{-...
Marc Palm's user avatar
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16 votes
2 answers
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Analogues of the Riemann-Roch Theorem

In his thesis, Tate derives a Poisson formula on the adeles and a theorem which he calls the "Riemann-Roch Theorem". More specifically, given a continuous, $L^1$ function $f$ on the adeles such that ...
Larry Rolen's user avatar
15 votes
2 answers
791 views

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 ...
JBorger's user avatar
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14 votes
3 answers
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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}} , \...
Andreas Holmstrom's user avatar
14 votes
3 answers
960 views

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, $|A(\mathbf{F}_{q^n})|$ ...
user avatar
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}...
H A Helfgott's user avatar
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12 votes
1 answer
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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) ...
Jeff Harvey's user avatar
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11 votes
2 answers
723 views

Tauberian theorem with better error term

This is a fairly vague question. Suppose we have a sequence of positive numbers $(c_n)_n$ and we want to find an asymptotic formula for $S(x) = \sum_{n \leq X} c_n$. In favorable circumstances, ...
Ramin's user avatar
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10 votes
1 answer
651 views

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 ...
Qiaochu Yuan's user avatar
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 ...
Eren Mehmet Kiral's user avatar
8 votes
1 answer
458 views

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 ...
Asvin's user avatar
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8 votes
2 answers
3k views

What is the Beilinson regulator?

Trying to understand answer to this question. What is the (Beilinson) higher regulator of a number field?
Ilya Nikokoshev's user avatar
8 votes
2 answers
803 views

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 $F$. The ...
Borp's user avatar
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7 votes
2 answers
792 views

Power series expansions of $L$-series

Let $\zeta_K(s)$ be the Dedekind zeta function for a number field $K$. We can understand the first non-vanishing coefficient of its Laurent series via the class number formula. Is anything known/...
flor.ian sprung's user avatar
7 votes
1 answer
365 views

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{...
QGravity's user avatar
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6 votes
2 answers
1k views

A (likely) positivity property of the Lerch zeta-function

The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where $$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$ is the Lerch zeta-...
Iosif Pinelis's user avatar
5 votes
6 answers
2k views

Sequences equidistributed modulo 1

Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results. H. Weyl (1909): The fractional part of the sequence $\alpha n$ is equidistributed modulo 1. I....
Nilotpal Kanti Sinha's user avatar
5 votes
1 answer
373 views

Results in an article by Siegel

Studying the Eisenstein cocycle by Sczech, I noticed that to understand its connection with the values at negative integers with zeta functions it is necessary to understand the resuts by Siegel in ...
efs's user avatar
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5 votes
0 answers
231 views

Orders of Clifford algebra

Let $C_n$ be the Clifford algebra over $\mathbb{Q}$ associated to negative definite quadratic form $-I_n$ (i.e. $-x_1^2-\dots-x_n^2$). Let $\mathcal{O}$ be a $\mathbb{Z}$-order of $C_n$. Q1) Is it ...
Subhajit Jana's user avatar
4 votes
0 answers
144 views

Ihara zeta function and closed paths and trails

Let $\Gamma$ be a finite graph. There seem to be two definitions of closed path in the literature. In one, a closed path is just a walk whose starting vertex is the same as the ending vertex. (Let us ...
H A Helfgott's user avatar
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3 votes
1 answer
688 views

Special values of Hecke L-function

The Dedekind zeta function for a number field $K$ is defined as $\zeta_K(s)=\sum_{I\subset O_K} (N_{K/\mathbb{Q}}(I))^{-s}$. By attaching a Hecke character $\psi$, we can define $L(s,\psi)=\sum_{I\...
Ted Mao's user avatar
  • 443
3 votes
2 answers
993 views

Castelnuovo Positivity (Rewrite of: Weil's original proof for FP^2)

Weil's proof of the Riemann Hypothesis for projective curves relies upon the following positivity result: Let $\mathbb{F}q$ be the finite field with $q$ elements, $\overline{\mathbb{F}q}$ its closure, ...
John McCarthy's user avatar
3 votes
1 answer
897 views

Is the integral always nonzero?

Let $$I_{n,p,a}:=\int_0^{\infty-} \frac{g_{n,a}(t)}{t^{p+1}} \, dt,$$ where $$(*)\qquad\qquad\qquad n\in\mathbb N,\quad -\infty<a<\infty,\quad p_{n,a} < p < n,\qquad\qquad\qquad\qquad\...
Iosif Pinelis's user avatar
3 votes
0 answers
434 views

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. ...
draks ...'s user avatar
  • 457
2 votes
1 answer
414 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/...
D.R.'s user avatar
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1 vote
0 answers
265 views

Does the property (P) holds true for the derivatives of $L$?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. As $s$ takes on real negative values, there are trivial zeros ...
China-Hong Kong's user avatar