14 problems

Let $Z_X(s)$ be the truncated Hadamard factor, let $X,T\to\infty$ with $X=O((\log T)^{2-\epsilon})$, and let $k>-1/2$ be fixed. Let $G$ be the Barnes $G$-function and let $\gamma$…

Let the non-trivial zeros of the Riemann zeta function be written as $\rho=\tfrac12+i\gamma$. Assume the Riemann Hypothesis and that these zeros are simple. Ng's conjecture. As…

Let $\lambda_j=\eta_j^2$ be the transformed nontrivial zeta zeros, ordered as in the source. For a parameter $\gamma$ with … consider consecutive transformed zeros that are not con…

Let $\rho_k$ be the nontrivial zeros of the Riemann zeta function with positive imaginary part, and set … Let $\Lambda$ be the associated entire function, and let its spacing const…

Let $K$ be a positive integer, let $a>0$, and consider the logarithmic derivative of the Riemann zeta-function at $1/2+a/\log T+it$. Near-critical logarithmic-derivative moment con…

Let $Z(t)$ be Hardy's function, so that $|Z(t)|=|\zeta(1/2+it)|$, and let $\sigma_{I\!I\!I'}$ be the function defined by the Painlevé equation referenced in the source. For non-neg…

Let $\psi(x)=\sum_{p^k\leq x}\log p^k$ be the weighted prime-counting function. Assume the Riemann hypothesis, and let the nontrivial zeros of the zeta function be written as…

Monotonicity conjecture. The function $F(s)$ is strictly increasing on $(1,\infty)$. In particular, $s=2$ is the unique zero of $F(s)$.

Soundararajan's conjecture. The following two statements are equivalent:

Let $\rho$ range over the non-trivial zeros of $\zeta(s)$, let $N(T)$ count zeros with $0<\operatorname{Im}(\rho)\leq T$, and let $P_X(s)$ and $Z_X(s)$ denote the Euler and zero fa…

Assume the Riemann Hypothesis. Let $\rho$ range over the non-trivial zeros of $\zeta(s)$, let $N(T)$ count those zeros with $0<\operatorname{Im}(\rho)\leq T$, and let $Z_X(s)$ be t…

Rightward correspondence conjecture. For $k\in\mathbb{N}$ there is a one-to-one correspondence between the zeros of $\zeta^{(k)}(s)$ and $\zeta^{(k+1)}(s)$, where the zero of…

Asymptotic periodicity conjecture. For all $\varepsilon>0$ and natural numbers $M$ and $N\geq 2$ there is $K\in\mathbb{N}$ such that for all $k>K$ there exists $s\in\mathbb{C}$ and…

Let the zeros of the Riemann zeta function be the complex numbers $\rho$ satisfying $\zeta(\rho)=0$. A single-process conjecture. The zeros of the $\zeta$ function are derived from…