19 problems

The Riemann zeta function is defined by $[... ELLIPSIZATION...$ and continued meromorphically to the complex plane. For complex shifts $[... ELLIPSIZATION...$, define the $2k$-poin…

Let $D$ and $K$ be nonnegative integers, and define the average moment of the $D$th derivative of the Riemann zeta-function on the critical line by … The random-matrix calculation…

Let $Z_F$ be constructed from the non-trivial zeta zeros, and consider orderings of those zeros, including the standard ordering by increasing imaginary part. Canonical Ordering co…

Let $Z_F$ be the fractal zero set constructed from the positive imaginary parts of the non-trivial zeros of $\zeta(s)$ by the recursive four-subinterval construction described in t…

Let $P_{\text{ess}}$ be the essential fractal prime set, with information measure $\iota(P_{\text{ess}})=-\zeta(1/2)$, and let $Z_F$ be the fractal zero set constructed from the po…

Derivative moment conjecture. As $\sigma\to\frac12$, one has

Logarithmic sink-proportion conjecture. The proportion of sinks versus nontrivial zeros follows a logarithmic distribution of the form

Source–sink infinitude conjecture. There exist infinitely many sources and infinitely many sinks on the critical line.

Let $T$ tend to infinity and consider the values of the Riemann zeta function on unit intervals along the critical line. Averaged local moment conjecture. … This conjecture is clos…

Let $T>1$, let $\tau$ be chosen uniformly from $[T,2T]$, and let $0<\alpha<1$ with $\theta\sim(\log\log T)^{-\alpha}$. Consider an interval of size … The smoothing-scale conjecture…

Let $T>1$, let $\tau$ be chosen uniformly from $[T,2T]$, and fix $\theta>0$. Consider the maximum of the Riemann zeta function over the interval $|h|\leq(\log T)^\theta$. The Argui…

Local ratios conjecture. For these fixed parameters,

GUE conjecture. Assume the Riemann hypothesis. For every fixed $k$ and every such test function $\eta$,

Let the normalized zeros of the Riemann zeta function be tested through their local correlation statistics. Montgomery's unrestricted-test-function conjecture. These correlations s…

Banks–Kang conjecture. This equation has no more than two solutions $t\in{\mathbb R}$ for every nonzero $z\in{\mathbb C}$.

Let $\zeta(s)$ be the Riemann zeta function, and consider the positive imaginary parts of its nontrivial zeros. The zeta linear-independence conjecture. These imaginary parts are l…

Let $B$ denote the Bernoulli operator, defined formally by $e^{-Bx}=\frac{x}{e^x-1}$, and let $\xi$ be the Riemann xi function. The expression $\sin \pi B\cdot\xi(B+s)$ is defined…

Let $zeta(s)$ be the Riemann zeta function, and consider the distribution of consecutive zeros of $zeta(s)$. Let … where $rho$ is a non-trivial zero and…

Let $\rho$ be a non-trivial zero of the Riemann zeta function $\zeta(s)$. Riemann's conjecture. All non-trivial zeros $\rho$ have real part equal to one-half, that is,…