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3
votes
2answers
407 views

Can we decide if an abelian variety is simple by knowing its Zeta function ?

Let $A$ be an Abelian variety defined over the finite field with $q$ elements. Let $P_i(T)$ be the characteristic polynomial of the action of the Frobenius on the $i^{th}$ étale cohomology group. Is ...
10
votes
2answers
903 views

Dedekind Zeta function: behaviour at 1

Let $\zeta$ denote the Dedekind zeta function of a number field $F$. We have $\zeta(s) = \frac{\lambda_{-1}}{s-1} + \lambda_0 + \dots$ for $s-1$ small. Class number formula: We have $\lambda_{-1} = ...
3
votes
0answers
212 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} ...
4
votes
1answer
333 views

Persistent homology of Markovian dynamical systems

Consider a dynamical system $(T,X)$ that admits a Markov partition $\mathcal{M}$ (e.g., an Anosov map), and consider the corresponding 0-1 transition matrix $A$. It is commonplace to study information ...
4
votes
1answer
249 views

Convergence of the summation 1/p^(1+iy) (over all primes p with y a nonzero real number)

For $z\in\mathbb{C}$ with real part greater than $1$ the sum $$\sum_{p}{\frac{1}{p^z}},$$ where the sum is taken over all primes $p$, converges absolutely. It is also well known that the same sum with ...
5
votes
1answer
411 views

Ihara zeta function

Is there a natural connection between the Ihara zeta function of a graph, and (for instance) the Riemann zeta function of certain varieties over finite fields ? Thanks.
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 ...
1
vote
0answers
150 views

reference-request for connection between numerator of zeta-function and characteristic polynomial of Frobenius on hyperelliptic curves over finite fields

Let $H$ be a hyperellipitic curve of genus $g$ defined over $\mathbb{F}_q$. The Frobenius endomorphism operates on the divisor class group of $H$ and satisfies a characteristic polynomial ...
4
votes
2answers
447 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 ...
7
votes
5answers
2k 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
3answers
1k views

Inverse of the Riemann zeta function

I'm wondering if there is any information on the inverse of the Riemann zeta function (not it's reciprocal, but its functional inverse). This would obviously be a multi-valued function.
3
votes
0answers
283 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 ...
1
vote
0answers
217 views

(why) Are the following two constructions of zeta functions equal?

Let $X$ be a variety defined over $\mathbb{Q}$. One has the usual Hasse-Weil zeta function. Now, let's do a different construction. Base change $X$ to $\mathbb{C}$: $X_{\mathbb{C}}$. Now look at its ...
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 ...
25
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
512 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
874 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
784 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
729 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 ...
8
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}} , ...
13
votes
5answers
2k 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: ...
13
votes
5answers
3k views

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 ...
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 ...
10
votes
1answer
776 views

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) ...
25
votes
1answer
2k views

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 ...
26
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 ...
12
votes
8answers
2k 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 ...
9
votes
1answer
1k views

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 ...
24
votes
2answers
1k 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 ...
2
votes
1answer
967 views

Function zeros in strip 0 < Re < 1 [closed]

Hi everyone. Could you plz tell me where the zeros of $f(s)$ in the strip $\{0 < \Re s < 1 \}$ are ? Do they all have $\Re s= 1/2$ ? $$f(s) = 1 - 2^{-s} - 3^{-s} + 4^{-s} - 5^{-s} + ...$$ ...
4
votes
1answer
239 views

Calculate the zeta function of a scheme by from its étale covers?

The title says it all. Given a (proper smooth) scheme $X$ over Spec$\mathbb{F}_q$, is it possible to calculate the zeta function of $X$ by from its étale covers? Like for $\mathbb{P}^n$ you can ...
14
votes
3answers
1k 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 ...
2
votes
1answer
528 views

Connection between Bernoulli polynomials and polygamma function

There is an intricate connection between Hurwitz Zeta and the (traditional) polygamma function: $$\psi_n(z)=(-1)^{n+1}n!\zeta(n+1,z)$$ If to use a generalization for Bernoulli numbers, this can be ...
6
votes
2answers
476 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
459 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 ...
7
votes
4answers
1k views

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, ...
10
votes
1answer
1k 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 $\zeta(2n+1) \notin (2\pi )^{2n+1} \mathbb{Q}$. I always assumed this was well known. More precisely I thought this result ...
14
votes
1answer
810 views

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 ...
10
votes
3answers
2k views

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 ...
5
votes
4answers
2k views

On partial sums of the zeta function, or $1/1 + 1/2 + 1/3 + … + 1/n$

When I was taking a shower this problem came into my mind... Let $f(n, s) = 1^{-s} + 2^{-s} + 3^{-s} + \cdots + n^{-s}$ be the partial sum of the $\zeta$ function. In the cases where $s$ is a ...
5
votes
1answer
477 views

Residues of $1/\zeta$

Are there any bounds on residues of $1/\zeta$ in roots of $\zeta$ in critical strip, which may use RH, but do not use the conjecture on simplicity of roots or something similar? I did not find such ...
3
votes
0answers
387 views

How looks the “land of Tamagawa numbers”?

Jonah Sinick's question here, other interesting ideas he mentioned, and Franz Lemmermeyer's remark make one think at Bloch and Kato's drawing + question. What's known or guessed about that "land" by ...
11
votes
1answer
605 views

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 ...
4
votes
1answer
623 views

2d Weil conjecture

Does there exist a two variable analogue of the Weil conjecture? What I mean is that the usual Weil involves a one-variable zeta-function which you get by using numbers $V_n = V ( GF(p^n))$ of points ...
5
votes
1answer
599 views

bibl. q.s on Dwork's “p-adic cycles”, Mazur's “p-adic variations”:

Matthew Emerton mentioned recently the relevance of Dwork's "p-adic cycles". As I wonder if I should read that, reviews of it are ambiguous, I'd be happy on remarks and possible further bibl. hints. ...
12
votes
3answers
2k views

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 ...
1
vote
0answers
386 views

Has the Weierstass transform been used to give Hermite series representations of the Riemann zeta function?

The inverse of the Weierstrass transform expands a function as a series of Hermite polynomials $H_{n}$. There are several ways to invert the Weierstrass transform which led me to the following ...
10
votes
1answer
702 views

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 ...