68 problems

Let $X^{(n)}_1$ be the number of distinct adjacent pairs of identical letters in a geometrically distributed word of length $n$, and let … where the independent random variables…

Conrey–Farmer–Keating–Rubinstein–Snaith conjecture. For some $\delta>0$,

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 $\gamma$…

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

Let $q\not\equiv2\pmod4$, and let $\varphi^*(q)$ denote the number of primitive characters modulo $q$. For each primitive character $\chi$ modulo $q$, write the non-trivial zeros o…

Let $\chi$ be a primitive Dirichlet character modulo $q>1$, and let $L(s,\chi)$ be its Dirichlet $L$-function, with non-trivial zeros $\rho=\tfrac12+i\gamma$. Assume the Generalise…

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…

Dirichlet $L$-function moment conjecture. For all $r\in\mathbb{C}$ with $\Re(r)>-3/2$, there is a constant $C_{q,r}$ such that

Let $\beta>0$, $k\in\mathbb{N}$, and $s\in\mathbb{R}_+$. Let $\mathsf{M}^{(\beta)}_N(k;s)$ denote the corresponding C$\beta$E field moment, and let $\mathfrak{c}^{(\beta)}(k;s)$ be…

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

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

Let $K$ be a number field containing the $\ell$-th roots of unity, let $\mathcal{O}_K$ be its ring of integers, and let $\mathcal{F}_\ell$ be a family of Hecke characters of order…

Let $\ell\geq1$ and $r\geq1$ be integers, and set … For the forms $f\in H_k(1)$, let $\lambda_f(\ell)$ denote the Hecke eigenvalue, let $L(s,f)$ be the associated $L$-function, and…

Effective non-vanishing conjecture. One has

Blomer–Khan–Young's fourth-moment conjecture. Under the normalization

Second-moment constant conjecture. One has

Let $k\in\mathbb{N}$ and let $0<\alpha\leqslant 1$ be irrational. Write $\zeta(s,\alpha)$ for the Hurwitz zeta function and let $d$ denote the algebraic degree of an algebraic numb…

Twisted fourth-moment hypothesis. For every coefficient sequence $\boldsymbol{\alpha}$ and every $\varepsilon>0$, the displayed estimate holds. This conjecture asks for the average…

Let $K$ be a regular $n$-sided polygon with $n\geq 3$, circumscribed by a circle of unit radius. For a positive even integer $p$, let $L_{22}^{(p)}$ denote the corresponding…

Let $a_{\mathrm{cusp}}(\sigma,n)$ be the cuspidal ideal-class counting function, let $\sigma\in N$ be an ideal class, and let $\beta>0$ be real. Theorem gives the order of growth o…

Let $a(\beta,\beta')$ denote the quantities appearing in Theorem, and let $\beta>0$ be real. For $\beta$ integral or half-integral, the corresponding $\beta$-moment has matching lo…

High-moment asymptotic conjecture. Under these conditions,

Let $\EuScript F_{\EuScript D}$ be the family of real primitive quadratic Dirichlet characters $\chi_d$, with $d$ nonzero and squarefree, and let $r\geq4$. For integers $N\geq1$ an…

Let $g$ and $h$ be admissible functions on $\mathbb{T}^k\times[0,1]^l$, with admissibility as in the xz-conjecture. Mathieu xz-conjecture. If … for every positive integer $P$, then…

Let $m_1,m_2,m_3\in\mathbb{N}$, and let $(X_1,X_2,X_3)$ be a centered Gaussian random vector. Three-dimensional Gaussian product inequality conjecture. … The equality sign is asser…