The zeta-functions tag has no usage guidance.

**49**

votes

**4**answers

3k 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 ...

**0**

votes

**1**answer

168 views

### This might be a trivial question on Hurwitz's zeta function.

In the book I am reading they write that for Hurwitz zeta function, $\zeta(x,s)=\sum_{n=0}^{\infty} \frac{1}{(x+n)^s}$, the next sum in the RHS converges for $\Re(s)>-1$, and I don't see how ...

**2**

votes

**1**answer

227 views

### The Hasse-Weil L-function and some equations

Let $f$ be an analytic function verfifying
$f(s)=\epsilon f(2-s)$
where $\epsilon=\pm 1$. The expression of Hasse-Weil L-function $f$ is
...

**21**

votes

**4**answers

2k 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 "exceptional zeros" of course first ...

**6**

votes

**2**answers

684 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 ?

**1**

vote

**0**answers

92 views

### binomial transform, Hurwitz zeta function

For $j,n\in\mathbb Z_+$,
let
$$
L_{j,n}^{(t)}=
\sum_{m=0}^{n} \Bigl(-\frac 12\Bigr)^{n-m}{n\choose m}{m+j+1\choose m+1} \left(
\frac {1}{t+\frac 12}\right)^{m+j+2}
$$
and
$$
L_{j,n} ...

**1**

vote

**2**answers

243 views

### Ihara zeta function (graph theory) coefficients using a line graph [closed]

I'VE COMPLETELY REVISED MY QUESTION
I wish to take a simple undirected graph (i.e. the complete graph K_4)
Arbitrarily direct said graph, and then create a line graph from the directed version of ...

**0**

votes

**0**answers

226 views

### Analytical continuation of the reciprocal of the Zeta function

Is the reciprocal of the Zeta function analytically continuable?
As $1/\zeta(n) = \sum_{n=1}^\infty \mu(n)/{n^s}$, this does not look obvious.

**0**

votes

**1**answer

120 views

### Are potential complex zeros not on the critical line of Dedekind zeta function in quadruples?

This question arose from sums over zeros of Dedekind zeta function.
It is known that complex zeros of Dedekind zeta function are in pairs $\rho, 1 - \rho$.
Is it true that potential complex ...

**4**

votes

**0**answers

307 views

### Does the Riemann hypothesis for liftable varieties over a finite field imply the Riemann hypothesis for all varieties over a finite field

The Riemann hypothesis for varieties over a finite field has been proven by Deligne. Still I would like to ask the following question.
A variety $X$ over a finite field $k$ is liftable if there ...

**1**

vote

**0**answers

139 views

### Special values of zeta functions and extensions of base fields.

Let $X$ be a scheme of finite type over a finite field $k=\mathbb{F}_{q}$ of $q$ elements.
Then, one can define the zeta function $Z_{X/k}(T)$ of $X$ ovet $k$ as $\prod_{x\in ...

**0**

votes

**2**answers

320 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 ...

**0**

votes

**2**answers

516 views

### Hadamard's product formula for the derivative

Let $f$ be an entire function of order $ρ<\infty$. Assume that $f$ does not vanish identically on $\mathbb{C}$. Then, we know that $f$ has a Hadamard's product formula
$$ f(s) =e^{g(s)}s^{r}\prod ...

**1**

vote

**1**answer

261 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 ...

**-11**

votes

**2**answers

850 views

### Riemann Siegel function and gamma function

I ask about an idea to prove this formula:
$Γ(1/2-iβ)=((\sqrt{π})/(\sqrt{\coshπβ}))\exp(-i(2ϑ(β)+βln2π+\arctan(\tanh(1/2)πβ)))$
where $ϑ(β)$ is the Riemann Siegel function.

**7**

votes

**1**answer

1k views

### 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 ...

**11**

votes

**1**answer

328 views

### 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:
...

**0**

votes

**1**answer

229 views

### Functional equation and constant functions

I ask about this claim:
let $f$ be an entire function satisfying $f(s)=u(s)f(a-s)$. Assume that $s$ and $a-s$ are not zeroes of $f$ and $f (bar)(a-s)=f(s)$ in a region $D$ ($f(bar)$ is the conjugate ...

**0**

votes

**0**answers

242 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

**1**answer

383 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

**2**answers

512 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

**1**answer

227 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

844 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 ...

**7**

votes

**0**answers

297 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 ...

**5**

votes

**2**answers

723 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

**4**answers

648 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 ...

**6**

votes

**3**answers

601 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

**2**answers

963 views

### Is there a known formula for fractional derivative of cot x?

I wonder if there any established formula for fractional derivative of a function $\pi \cot (\pi x)$.
I derived the following expression:
$(\pi \cot (\pi ...

**10**

votes

**2**answers

927 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 ...

**5**

votes

**1**answer

432 views

### Is Zeta function discrete-analytic?

Let's define discrete-analytic functions as functions that are equal to their Newton series expansion:
$$f(x) = \sum_{k=0}^\infty \binom{x-a}k \Delta^k f(a)$$
My question is whether $\zeta(s,q)$ ...

**8**

votes

**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 ...

**3**

votes

**0**answers

822 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 ...

**3**

votes

**2**answers

429 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 ...

**14**

votes

**2**answers

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

**0**answers

231 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

**1**answer

437 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

**1**answer

258 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

**1**answer

459 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.

**28**

votes

**4**answers

4k 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

**0**answers

172 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

**2**answers

486 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 ...

**8**

votes

**5**answers

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 ...

**3**

votes

**3**answers

2k 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

**0**answers

307 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

**0**answers

242 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 ...

**48**

votes

**3**answers

5k 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 ...

**33**

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**3**answers

3k 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

**1**answer

557 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 ...

**15**

votes

**2**answers

1k 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

885 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. ...