7
votes
1answer
247 views
Best bounds toward Serre’s uniformity conjecture
If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre
that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the
Galois r …
32
votes
3answers
1k 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. Ar …
12
votes
3answers
695 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" …
3
votes
2answers
227 views
Mertens function limits using $\phi+2$n
As we can see in the plot below, Mertens function: $M(x)\equiv \sum_{n=1}^{x}\mu(n)$ has wild swings from positive to negative and back again.
When we use: $$x=\frac{1}{2+\fra …
4
votes
2answers
267 views
Equivalence of two well-known forms of (RH): reference-request.
This is a reference-request about a very simple statement.
The Riemann hypothesis is well-known to be equivalent to
$$(1)\ \ \ \pi(x) = \mathrm{Li}(x)+O(x^{1/2} \log x)$$
and to …
4
votes
1answer
246 views
Is there a connection between the closed forms of these two infinite products?
Take the following two infinite products that have closed forms.
Assume: $\gamma_n > 0 \in \mathbb{R};s,a,x \in \mathbb{C}; x \ne 0,a \pm ix\gamma_n \ne 0$
The first product:
$ …
4
votes
3answers
411 views
A closed form of infinite products of complex zeros involving $\Im(\rho_n)$. Does a proof of this closed form imply RH?
Building on this question scaling the imaginary part of $\rho$s in infinite products, I like to conjecture that:
$$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\mu_n} \right …
2
votes
2answers
167 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 …
13
votes
1answer
728 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 …
4
votes
4answers
411 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, $s \in \mathbb{C}$:
$$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\mu} \right) \left(1- \fra …
2
votes
1answer
730 views
What happens when infinite values of $\zeta_{H}(s,z)$ approach $\zeta(s)$ ?
Take the following Hurwitz zeta:
$$\zeta_{H}(s,z)$$
with $s=\sigma \pm ti$ and $\displaystyle z=1 \pm \frac{i}{a}$ and $t,a \in \mathbb{R}$.
In the critical strip $0 \lt \sigma …
14
votes
1answer
705 views
A question about Speiser’s 1934 result on the Riemann hypothesis
A number of sources concerning Speiser's 1934 result state that the Riemann Hypothesis (RH) implies $\zeta'(s)\neq 0$ for all $0<\text{Re}(s)<1/2$. But I have seen some (poss …
1
vote
1answer
228 views
$\zeta(2k+1)$ expressed in a product of two infinite products of non-trivial zeros.
Take the Hadamard product for $\zeta(s)$:
$$\displaystyle \zeta(s) = \pi^{\frac{s}{2}} \dfrac{\prod_\rho \left(1- \frac{s}{\rho} \right) \left(1- \frac{s}{1-\rho} \right)}{2(s-1)\ …
0
votes
0answers
341 views
What Fields of Graduate Study to Master Riemann Hypothesis [closed]
I'm thinking what are the specific PG courses / areas of Maths I need to take to become an expert on the Riemann Hypothesis?
1
vote
0answers
98 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 …

