65 problems

Generalized Xi identity conjecture. For all $s_0\in\mathbb C$ and all $a\in\mathbb C$,

Let $k$ be a positive integer, let $\theta(t)$ be the Riemann–Siegel theta function, and define Hardy's function by $Z(t)=e^{i\theta(t)}\zeta(1/2+it)$. Let $a_k$ be the arithmetic f…

Let $Z_X(s)$ denote the truncated zeta-function used in the paper, and let $A>0$ be fixed. Truncated-zeta maximum conjecture. If $2<X<\log^A T$, then, as $T\to\infty$, … The conjec…

Let $S(t)$ denote the error term in the number of zeros of the Riemann zeta-function up to height $t$. Limsup conjecture. … The conjecture is motivated by arguments analogous to th…

Assume the Riemann Hypothesis and let $\gamma,\gamma'$ range over imaginary parts of non-trivial zeros of $\zeta(s)$. For $0<\alpha<\beta\ll\log T$, the short-interval pair-correla…

Assume the Riemann Hypothesis, let $\gamma$ and $\gamma'$ range over imaginary parts of non-trivial zeros of $\zeta(s)$, and let $h$ be the shift parameter. For fixed $\alpha>0$, t…

Let $N(T)$ denote the number of non-trivial zeros $\tfrac12+{\mathrm{i}}\gamma_n$ with $0<\gamma_n\leq T$, and set … Let $F_2$ be the function specified by the paper's formula for…

Let $N(T)$ denote the number of non-trivial zeros $\tfrac12+{\mathrm{i}}\gamma_n$ with $0<\gamma_n\leq T$, and set … For $k$ with $\operatorname{Re}(k)>-1/2$, let $a(k)$ be the ari…

Gonek–Hejhal conjecture at $k=1$. One has

Keating–Snaith moment conjecture. For every such $\boldsymbol{\kappa}$,

Linear Independence Conjecture. This set is linearly independent over $\mathbb{Q}$. The conjecture is used in the paper's large-deviation arguments for the summatory functions unde…

Let $\kappa$ parametrize the operators $D_{\log}^{(\kappa,\kappa)}$, and let $\mathcal{E}(\kappa)$ denote the numerical error between their eigenvalues and the imaginary parts of t…

Hughes–Keating–O'Connell conjecture. For any fixed $k$ with $\Re(k)>-3/2$,

Hughes–Pearce-Crump conjecture. For $\Re(k)>-3$,

Let $N^*(T)$ be the multiplicity-weighted zero count … where $m_\rho$ is the multiplicity of the zero $\rho$. Montgomery's simplicity conjecture. As $T\to\infty$, … Since…

Let $\rho'=\beta'+i\gamma'$ be a sequence of type 2 zeros of $\zeta'(s)$, and let the sum range over the other zeros $\lambda'\ne\rho'$ of $\zeta'(s)$. Stopple's type 2 curvature e…

Let a Z-curve be one of the level curves described in the source for $\zeta'(s)$, and classify zeros of $\zeta'(s)$ by types $0$, $1$, and $2$. Write a zero as…

Assume the notation $N_0'(T)$, $N_1(T)$, and $N_2(T)$ denotes the counting functions up to height $T$ for zeros of $\zeta'(s)$ of types $0$, $1$, and $2$, respectively. Stopple's z…

Conjectures on extreme values. One may conjecture that the number of extreme values up to height $T$ is less than $C\log T$ for some constant $C$, and that the sets of indices $n$…

Let $\theta\in(-1,0]$, let $T$ tend to infinity, and let $\tau$ be uniformly distributed on $[T,2T]$. Write $\mathcal{N}(0,1)$ for a standard Gaussian random variable, and let…

Fix $\sigma\in(\frac12,1)$, and let $\zeta(s)$ denote the Riemann zeta function. Pointwise upper-bound conjecture. For all sufficiently large $t$, … The source lists this as a conj…

Fix $\epsilon\in(0,\frac12)$. For $T>0$, consider the interval $[T,2T]$, and let $\zeta(s)$ denote the Riemann zeta function. Directional maximum conjecture. As $T\to\infty$, … uni…

Effective Linear Independence conjecture. The above lower bound holds for all such $T$, $\varepsilon$, and integer coefficients. This is an effective quantitative strengthening of…

Let $a,k,\beta\in(0,\infty)$ and let $\sigma\in(\frac{1}{2},1]$ be fixed. Mixed zeta-ratio moment conjecture. There is a constant ${\mathcal D}(\sigma,a,k,\beta)$, depending only o…

Smoothed weighted one-level density conjecture. As $T\to\infty$,