# Tagged Questions

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vote

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213 views

### prime zeta function when $0<s<1$

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 ...

**1**

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**0**answers

159 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 ...

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208 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 ...

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**1**answer

225 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}
...

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**1**answer

513 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**

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**0**answers

237 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**

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**0**answers

192 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 ...

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**6**answers

749 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.
...

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415 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 ...

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**1**answer

343 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 ...

**36**

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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 ...

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**4**answers

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 ...

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**2**answers

600 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**

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**2**answers

275 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 ...

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**1**answer

223 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 ...

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**0**answers

215 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 ...

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**1**answer

366 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.

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**2**answers

456 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 ...

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**1**answer

204 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

**1**answer

836 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**

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**0**answers

263 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 ...

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610 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 ...

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**4**answers

562 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 ...

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**3**answers

531 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 ...

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**2**answers

613 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 ...

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**3**answers

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 ...

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707 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 ...

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902 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**

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**0**answers

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} ...

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**4**answers

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 ...

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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 ...

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**0**answers

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 ...

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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 ...

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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**

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**1**answer

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 ...

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**2**answers

873 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

**2**answers

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. ...

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**1**answer

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 ...

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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 ...

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**3**answers

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}} , ...

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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 ...

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**3**answers

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: ...

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**1**answer

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 ...

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**3**answers

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 ...

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**3**answers

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 ...

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**2**answers

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**

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**1**answer

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**

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**4**answers

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, ...

**14**

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**1**answer

807 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 ...

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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 ...