**5**

votes

**1**answer

498 views

### Is there a connection between the closed forms of these two infinite products?

Take the following two infinite products that have closed forms.
Assume: $\gamma_n > 0 \in \mathbb{R};s,a,x \in \mathbb{C}; x \ne 0,a \pm ix\gamma_n \ne 0$
The first product:
$$\displaystyle ...

**5**

votes

**2**answers

312 views

### Equivalence of two well-known forms of (RH): reference-request.

This is a reference-request about a very simple statement.
The Riemann hypothesis is well-known to be equivalent to
$$(1)\ \ \ \pi(x) = \mathrm{Li}(x)+O(x^{1/2} \log x)$$
and to
$$(2)\ \ ...

**2**

votes

**2**answers

297 views

### On the location of zeros of L functions from modular forms

I understand that the Mellin transform of a modular form is expected to satisfy RH when it is an eigenform of all Hecke operators, in which case it has an Euler product. Now about when the form is not ...

**5**

votes

**3**answers

611 views

### A closed form of infinite products of complex zeros involving $\Im(\rho_n)$. Does a proof of this closed form imply RH?

Building on this question scaling the imaginary part of $\rho$s in infinite products, I like to conjecture that:
$$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\mu_n} \right) \left(1- ...

**14**

votes

**1**answer

960 views

### Is there a Montgomery's conjecture for Dirichlet characters and Artin representations ?

Edit: as GH noticed, the way I tried to state Montgomery's conjecture is wrong. There were some mistakes in the references I used, which compounded with some mistakes of mine, gave a very poor post. ...

**4**

votes

**4**answers

472 views

### What happens to $\zeta(s)$ when all its $\Im(\rho_n)$ are “scaled” linearly?

I found that the following infinite product with $\mu = a +n b i$ and a,b real, $s \in \mathbb{C}$:
$$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\mu} \right) \left(1- \frac{s}{1-\mu} ...

**1**

vote

**1**answer

351 views

### $\zeta(2k+1)$ expressed in a product of two infinite products of non-trivial zeros.

Take the Hadamard product for $\zeta(s)$:
$$\displaystyle \zeta(s) = \pi^{\frac{s}{2}} \dfrac{\prod_\rho \left(1- \frac{s}{\rho} \right) \left(1- \frac{s}{1-\rho} ...

**2**

votes

**0**answers

150 views

### The influence of $\chi(s)$ on complex zeros of $\frac{\zeta(\overline{s})}{\zeta(1-s)} \pm \frac{\zeta(s)}{\zeta(1-\overline{s})}$

I was exploring the formula:
$$g(s)_{\pm} := \displaystyle \frac{\zeta(\overline{s})}{\zeta(1-s)} \pm \frac{\zeta(s)}{\zeta(1-\overline{s})}$$
and found that for all $\Re(s) \ne \frac12$:
...

**15**

votes

**1**answer

896 views

### A question about Speiser's 1934 result on the Riemann hypothesis

A number of sources concerning Speiser's 1934 result state that the Riemann Hypothesis (RH) implies $\zeta'(s)\neq 0$ for all $0<\text{Re}(s)<1/2$. But I have seen some (possibly less reliable) ...

**0**

votes

**0**answers

243 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 ...

**0**

votes

**1**answer

681 views

### Interplay between Riemann and Swinnerton-Dyer

Hello everyone,
After reading RH ( Riemann's Hypothesis ) and Swinnerton-Dyer conjecture, I asked myself why can't RH hold for $L$-Functions ( Hasse-Weil L-function ). In particular the GRH imposes ...

**2**

votes

**1**answer

228 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) + ...

**8**

votes

**1**answer

1k views

### What is the relation between Quasicrystals, Riemann Hypothesis, and PV numbers?

Could somebody explain to me, from a mathematical stand-point, what is a quasi-crystal, and how it relates to the set of Pisot numbers, and the Riemann Hypothesis?
I've heard Freeman Dyson say that ...

**3**

votes

**1**answer

845 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 ...

**3**

votes

**0**answers

279 views

### Definite integral of $\zeta(s)$ over the critical strip

Take the following definite integral:
$$f(s):=\int_s^{1-s} \zeta(x) \mathrm{d} x$$
with $s \in \mathbb{C}$, $s=\sigma \pm ti$, $0<\sigma<1$ and $t,\sigma \in \mathbb{R}$.
The graph of ...

**3**

votes

**2**answers

444 views

### Are all zeros of ζ^{k}(s)±ζ^{k}(1−s) on the critical line (k=k-th derivative)?

The non-trivial zeros of $\zeta^{k}(s)$, with $k=k^{th}$ derivative, do not lie on a line and seem to be distributed randomly in the region $\sigma > \frac12$. However the non-real zeros in the ...

**1**

vote

**1**answer

483 views

### Zeros of the function $\zeta(s) \pm \zeta(\overline s)$

Building on this question: Zeros of $\zeta(s) \pm \zeta(1-s)$, I experimented further with:
$$\zeta(s) \pm \zeta(\overline s)$$
Assuming $s=\sigma + ti$, I observed that this function also has many ...

**10**

votes

**1**answer

652 views

### Montgomery's pair correlation function without RH?

In the theory of the Riemann zeta function, Montgomery's Pair correlation function is defined as
$$
F(\alpha) = \frac{1}{N(T)}
\sum_{T < \gamma, \gamma' < 2T} T^{i \alpha (\gamma - \gamma')} ...

**20**

votes

**5**answers

1k views

### Are the 'semi' trivial zeros of $\zeta(s) \pm \zeta(1-s)$ all on the critical line?

The proof that $\Gamma(z)\pm \Gamma(1-z)$ only has zeros for $z \in \mathbb{R}$ or $z= \frac12 +i \mathbb{R}$ has been given here:
Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real ...

**-1**

votes

**1**answer

571 views

### Is there information about the $\rho$'s hidden in the zeros of $\Re(\chi(s))$ ?

Take the symmetrical form of the completed Zeta-function:
$\displaystyle \chi(s) \zeta(s) = \chi(1-s) \zeta(1-s)$
with
$\chi(s)=\pi^{-(\frac{s}{2})} \Gamma(\frac{s}{2})$.
For $s=\sigma + ti$, I ...

**0**

votes

**3**answers

590 views

### Possible locations for non trivial zeroes lying off the critical line

It has been proven that:
1) if $s$ is a non trivial zero $\rho$ of $\zeta(s)$ then so is $1−s$.
2) $\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)$
3) $ 0 < \Re(\rho) ...

**9**

votes

**3**answers

1k views

### On Robin's criterion for RH [closed]

\begin{equation}
\sigma(n) < e^\gamma n \log \log n
\end{equation}
In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin ...

**3**

votes

**2**answers

410 views

### using distribution of primes to generate random bits?

In his popular science book The Music of the Primes, Marcus du Sautoy tries to link the truth of the Riemann Hypothesis to the "randomness" of the primes. To do this, he invokes the idea of a "fair ...

**21**

votes

**2**answers

1k views

### Given an integer polynomial, is there a small prime modulo which it has a root?

I am looking at a paper by Pascal Koiran on the computational complexity of certifying the solvability of integer polynomial equations in several variables. With the aid of some important theorems in ...

**2**

votes

**0**answers

366 views

### Characterizing essential singularities

In the paper Picture of an essential singularity, an analogy is made between the multipolar moments of infinitesimal charge distributions and the lines of constant modulus/argument around an essential ...

**13**

votes

**1**answer

1k views

### Exceptional zeros and Liouville's $\lambda$ function

This originated from an textbook exercise (recently posted to math.stackexchange
http://math.stackexchange.com/questions/62883/quadratic-characters-and-liouvilles-function
with no success) but I think ...

**4**

votes

**4**answers

2k views

### Which conjectures only need the Grand Riemann Hypothesis to become genuine theorems?

Hello,
I've been interested in number theory for several years, and as time goes by, I read more and more articles in which theorems begin with "Assume the Riemann Hypothesis holds." But up to now, I ...

**30**

votes

**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 ...

**70**

votes

**5**answers

25k views

### Consequences of the Riemann hypothesis

I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know?
It would also be nice ...

**22**

votes

**4**answers

3k views

### Modular forms and the Riemann Hypothesis

Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions?
What I'm thinking of is this: under the Mellin transform, the Riemann zeta function ...