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 …

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 …

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" …

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 …

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 …

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: $ …

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 …

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 …

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 …

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 …

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 …

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 …

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)\ …

I'm thinking what are the specific PG courses / areas of Maths I need to take to become an expert on the Riemann Hypothesis?

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 …