5 problems

Let $\rho$ range over the non-trivial zeros of the Riemann zeta function, let $\operatorname{N}(T)$ denote the number of zeros with $0<\operatorname{Im}(\rho)\leq T$, and let…

Let $\ell\in\mathbb N$ and $\sigma\in(0,1)$ be fixed. Comparison conjecture. There exist constants $C(\sigma,\ell)$ and $c(\sigma,\ell)$, depending on $\sigma$ and $\ell$, such tha…

Let $\ell\in\mathbb N$ be fixed, and write $\log_2 T=\log\log T$ and $\log_3 T=\log\log\log T$. Polynomial asymptotic conjecture. There exists a polynomial $P_{\ell+1}(x,y)$ of tot…

Positive-proportion conjecture. There exist positive constants $c_1$ and $c_2$, with

Let $\chi$ be a primitive Dirichlet character, and let $K$ be an Abelian number field. Write $\rho=\beta+i\gamma$ for nontrivial zeros of the relevant Dirichlet $L$-function…