9
votes
1answer
402 views

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 ...
2
votes
2answers
302 views

Local factors of Hasse-Weil zeta function - what do they have in common?

Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of ...
8
votes
2answers
428 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 $F$. ...
1
vote
2answers
237 views

prime zeta function when $0<s<1$ [closed]

I will not be surprised if this question seems trivial in MO but i asked it first in MathSE and i did not get an answer. So, here it is: I would like to know if there is a good estimate for the sum ...
2
votes
0answers
179 views

Automorphicity of L-Factors of Zeta Functions

Associated to a variety over a number field $K$, one has a family of ``Hasse--Weil'' L-functions, which can be combined (as an alternating product) to give the Hasse--Weil zeta function of the ...
10
votes
0answers
224 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 ...
6
votes
1answer
242 views

Gamma Factors for Zeta Functions of Abelian Varieties

If $X$ is a scheme of finite type over $\mathbb{Z}$, and $X_0$ denotes its set of closed points, then one can define its zeta function on the half plane $Re(s)>\text{dim}(X)$: \begin{equation} ...
1
vote
1answer
549 views

An integral representation of the Riemann zeta function

I am referring to the equality in equation $3.29$ (page 12) and $4.20$ (page 17) in this paper. I am unable to recognize where this comes from or what is the general expression for values other than ...
4
votes
0answers
238 views

Zeta functions for infinite abelian extensions?

Are there any reasonable ways to define zeta function of a maximal abelian extension $\mathbb{Q}^{ab}$ of the rationals, or at least zeta functions of some its infinite-dimensional subfields? In ...
2
votes
0answers
199 views

computing a certain contour integral [closed]

I want to compute an integral along a vertical line segment. The function I'm integrating involves the zeta-function, and usually the way such integrals are done treats the line segment as one side ...
6
votes
6answers
807 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. ...
1
vote
2answers
454 views

A generalisation of the Birch and Swinnerton-Dyer conjecture

We know that the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and Generalized Riemann hypothesis. My question is about the existence of a similar generalisation of the Birch ...
6
votes
1answer
356 views

Efficient (divergent) summation for sum of zetas at negative arguments?

In a question in MSE (see bottom of my own answer) I'm considering the following series, depending on a parameter m: $$ L(m) = -\zeta(1m)/1 - \zeta(2m)/2 - \zeta(3m)/3 - \ldots $$ where I want to make ...
46
votes
4answers
2k 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 ...
18
votes
4answers
1k views

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 ...
5
votes
2answers
631 views

Is there an explicit expression for the imaginary part of some non-trivial zero of zeta,

Is there an explicit expression for the imaginary part of some non-trivial zero of zeta, in terms of well-known constants, such as say $\gamma$ or $\pi$ say ?
0
votes
2answers
291 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: http://mathoverflow.net/questions/115179/real-root-1-of-the-hasse-weil-l-function-of-c-over I am just curious to know ...
1
vote
1answer
234 views

Question about the function $\zeta(s) \pm \dfrac{1}{\zeta(1-s)}$

It is easy to see that the function: $$\zeta(s) \pm \dfrac{1}{\zeta(1-s)}$$ has a pole at each non-trivial zero $s=\rho_n$. However, after some experiments with this function, I would like to ...
0
votes
0answers
218 views

Does there exist a Weierstrass/Hadamard factorization for $\chi(s)-1$ ?

Would like to build once more on this question. Take $s=\sigma + ti, s \in \mathbb{C}, 0<\Re(\sigma)<1$. Let's assume it is proven that: $$\zeta(1-s) - \zeta(s)$$ has all its zeros on the ...
1
vote
1answer
370 views

Functional equation of the alternating zeta function

Can one let me know about the functional equation of the alternating zeta function similar to the well known for the rieman function.
6
votes
2answers
467 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 ...
2
votes
1answer
221 views

Deriving the Riemann non-trivial zeros from $\zeta_{H}(s,a) + \zeta_{H}(s,1-a)$

The Hurwitz zeta function: $$\zeta_{H}(s,a)$$ reduces to $\zeta(s)$ when $a=1$ and to $(2^s-1)\zeta(s)$ when $a=\frac12$. However, I stumbled upon a peculiar third connection: $$\zeta_{H}(s,a) + ...
3
votes
1answer
838 views

What happens when infinite values of $\zeta_{H}(s,z)$ approach $\zeta(s)$ ?

Take the following Hurwitz zeta: $$\zeta_{H}(s,z)$$ with $s=\sigma \pm ti$ and $\displaystyle z=1 \pm \frac{i}{a}$ and $t,a \in \mathbb{R}$. In the critical strip $0 \lt \sigma \lt 1$, this Hurwitz ...
6
votes
0answers
272 views

Multiple zeta values at negative integers

For a tuple of integers $\underline{s}:=(s_1,\dots, s_d)$, the multiple zeta value (MZV) at $\underline{s}$ is defined as: $$ \zeta(s_1,\dots, s_d):= \sum_{n_1>\dots>n_d>0}\frac 1 ...
4
votes
2answers
656 views

why Riemann Hypothesis over curves is easy but 'normal' hypothesis for the Riemann Zeta function $ \zeta (s) $ is hard ,

despite both zetas $ \zeta (s,X) $ and $ \zeta (s)$ have the same functional equation, the same Euler prodcut and the same Riemann-Weil formula why one of them is 'easy' and can be solved but the ...
11
votes
4answers
593 views

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 ...
5
votes
3answers
556 views

A question about partial Euler products

Let $K/{\mathbb Q}$ be an extension of degree $d$. Let $S$ be the set of primes $p$ which split completely in $K$. What can one say about the analytic properties of $$ \zeta_{K, S}(s) : = \prod_{p ...
7
votes
2answers
658 views

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 ...
7
votes
3answers
2k views

Riemann zeta at even integers

I am talking about this in a course I am teaching, and hence am wondering: what are the various derivations of the values of Riemann zeta function at even integers? There are two incredibly cool ...
2
votes
0answers
730 views

Proof that derivative of Hurwitz Zeta by the first argument is not expressable in terms of Hurwitz Zeta

The set of elementary functions is defined so that it to be closed against operation of differentiation. It is also evidently close against discrete differentiation. In the discrete calculus there is ...
14
votes
2answers
1k 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 ...
3
votes
0answers
217 views

Alternating sums of powers of the lngamma ($\small f_p(x) = \sum \log(k!)^p x^k $ at $\small x=-1$)

I'm still fiddling with this recent question and come to a detail, whether I can find closed forms for the sums of the $\small lngamma() $ function. Precisely $$ \small f_p(x) = \sum_{k=0}^{\infty} ...
27
votes
4answers
3k 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 ...
4
votes
2answers
455 views

How do Brauer groups relate to zeta functions?

There are two approaches to class field theory that I was taught. The first, is the theory of $L$-functions, Dirichlet characters and so forth (which I described succintly in the question What are the ...
3
votes
0answers
288 views

Is there a notion of a zeta function of a morphism?

The Hasse-Weil zeta function is defined only for arithmetic schemes. By an arithmetic scheme I will mean a scheme $X$ together with a morphism of finite type $X\rightarrow S$, where $S$ is an affine ...
47
votes
3answers
4k views

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 ...
27
votes
3answers
2k 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) = ...
9
votes
1answer
519 views

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 ...
14
votes
2answers
901 views

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 ...
4
votes
2answers
811 views

Evaluating the integral $\int_0^\infty \frac{\psi(x)-x}{x^2}dx.$

Let $\psi(x)=\sum_{n\leq x} \Lambda(n)$ be the weighted prime counting function. I am trying to evaluate the integral $$\kappa:=\int_{1}^{\infty}\frac{\psi(x)-x}{x^{2}}dx$$ in several different ways. ...
13
votes
1answer
748 views

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 ...
16
votes
2answers
1k 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 ...
9
votes
3answers
1k views

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}} , ...
14
votes
5answers
3k 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 ...
17
votes
3answers
1k views

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: ...
12
votes
1answer
1k views

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 ...
27
votes
3answers
3k views

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 ...
15
votes
3answers
2k views

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 ...
6
votes
2answers
497 views

Poles of Kloosterman Zeta Function

I am a string theorist who has encountered the following number theory problem in my research. Consider the sums $$Z_+(s) = \sum_{p=1}^\infty p^{-s} S(1,1; p)$$ and $$Z_-(s) = \sum_{p=1}^\infty ...
4
votes
1answer
469 views

Characteristic Complexes in Iwasawa theory

For all that follows, $p$ is a fixed odd prime. In the formulation of the Noncommutative Main Conjecture of Iwasawa theory one uses étale cohomology to define an algebraic object analogous to Iwasawas ...