16 problems

Soundararajan's conjecture. The equality $\lambda=0$ holds if and only if $\lambda'=0$. This conjecture relates unusually small gaps between consecutive zeta zeros to zeros of…

Let $S$ denote the multiset of ordinates of the nontrivial zeros of the Riemann zeta function. For $a>0$, define the continuous function $H_a:\mathbb{R}_{>0}\to\mathbb{C}$ by … whe…

Limiting-distribution conjecture. For every $a>0$ there is a Borel function $f_a:\mathbb{C}\to\mathbb{R}$ satisfying

Let $F(x,T)$ be the weighted pair-correlation sum over imaginary parts $\gamma,\gamma'$ of non-trivial zeros of the Riemann zeta function, with … For any small $\epsilon>0$ and any…

Let $\rho'=\beta'+i\gamma'$ be a type 2 zero of $\zeta'(s)$, and let $\gamma^-$ and $\gamma^+$ denote the ordinates of the neighboring zeros of $\zeta(s)$ relevant to its type 2 cl…

For a positive integer $\mu$, let ${\mathfrak Z}_\mu(T)$ denote the set of $\mu$-tuples of zero ordinates under consideration, let…

Let $\gamma$ range over the positive ordinates of zeros of the Riemann zeta function. Linear independence conjecture. The positive ordinates of the zeros of the Riemann zeta functi…

Quantitative linear independence conjecture. There exists a constant $c_k>k$ such that, for every $\boldsymbol{\epsilon}\in\{\pm1\}^k$,

Ki's conjecture. The logarithmic derivative of the zeta-function satisfies

GUE hypothesis. For $n\geq 1$ and any $\varphi\in\mathcal{S}(\mathbb{R}^n)$,

Let $G(s)$ be the function considered above, and let $T$ tend to infinity. Mean-square conjecture. … The preceding theorem gives the unconditional bounds…

Let $N(T)$ denote the zero-counting function defined by (1.5), and let the sum range over zeros $\rho=\beta+i\gamma$ of the Riemann zeta-function with $0<\gamma\leq T$. For each in…

Derivative-zero density conjecture. If $m'(\nu)\gg\nu^\alpha$ for some $\alpha<2$, then $m(\nu)>0$ for all $\nu>0$. This is proposed as a refinement of Soundararajan's conjecture.…

Fujii's conjecture. These values are uniformly distributed modulo one in the sense of Weyl. The conjecture describes the distribution of the functional-equation factor evaluated at…

Let $(\beta_n+i\gamma_n)$ be the zeros of $\zeta(s)$ in the upper half-plane, arranged in nondecreasing order and counted according to multiplicity, with $\gamma_n\leq\gamma_{n+1}$…

Let the zeros of the Riemann zeta function be counted with their multiplicities. A zero is simple when its multiplicity is one. Simplicity conjecture. All or at least almost all ze…