732 problems

For every integer $k\ge 1$ and every collection of distinct natural numbers $h_1,\ldots,h_k$, where $\lambda(n)=(-1)^{\Omega(n)}$ is the Liouville function, one has…

For integers $r\ge 1$ and real numbers $n\ge 1$, $0<\eta\le 1/2$, define … Here $a_i,b_j\sim n$ denotes the same dyadic range as in the source, and $\lVert\cdot\rVert$ is distance…

Kionke's conjecture. The shifted partial Riemann zeta function $\zeta_S(s-1)$ has the same abscissa of convergence as the Weil representation zeta function $\zeta^W_{H_S}(s)$. This…

Let $\lambda$ denote the Liouville function, and let $H=\{h_1,\ldots,h_d\}$ be a non-empty set of non-negative integers. Chowla's conjecture. The correlation of the Liouville funct…

Refined moment conjecture. As $x\to\infty$,

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

Let $G_2(n)=\sum_{k+m=n}\Lambda(k)\Lambda(m)$ for positive integers $n$, where $\Lambda$ is the von Mangoldt function, and define … Under the Riemann hypothesis, $\Phi_2(s)$ admits…

The orthogonal sector variance conjecture. If $x\leq K^\ell$, then there is a constant $a^O_\ell\in\mathbb Q$, depending on $\ell$, such that

Let $d_k(n)$ denote the $k$-fold divisor function, and define … For $x^{1/k}\leq y$ and primes $p$ with $y\leq p\leq 2y$, write $\mathrm{Var}_{p\in[y,2y]}$ for the variance over th…

Let $X>1$, let $q$ be a positive integer, and let $a$ range over residue classes relatively prime to $q$. Define the error term $E(X,q,a)$ by … where … For $ell\geq 1$, define the…

For a natural number $h$, let $n_h$ be the smallest ternary integer, if it exists, such that $A(n_h)=h$. Polynomial-growth conjecture for the least ternary height representative. T…

Generalized $\tau$-Li criterion. Non-vanishing of $F$ in the half-plane $\operatorname{Re}(s)>\tau/2$ is equivalent to growth of $\lambda_F(n,\tau)$ as $\mathcal{C}_F\tau n\log n$…

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 $L$ be a sublattice of $\mathbb{Z}^t$, let $N$ be large, let $K\subseteq[-N,N]^t$ be convex with volume $\gg N^t$, and let $a\in\mathbb{Z}^t$ have size $O(N)$. Assume that all…

Let $L$ be an affine sublattice of $\mathbb{Z}^t$. For each prime $p$, let $\alpha_p(L)$ be the local factor defined from the image $L_p$ of $L$ in $\mathbb{Z}_p^t$; a local obstru…

Polylogarithmic prime-gap conjecture. There exists an absolute constant $k>1$ such that

Let $\theta$ satisfy $1/2<\theta\leq 1$. Let $q$ be a positive integer and let $c$ be an integer with $(c,q)=1$. Coprime-factor congruence conjecture in short intervals. There is a…

Let $\epsilon>0$ be any small positive real number, let $q$ be a positive integer, and let $c$ be an integer with $(c,q)=1$. Short-factor congruence conjecture. The congruence … ha…

For $\alpha>0$, let $A(\alpha)$ be the asymptotic constant conjectured for … and let $B(\alpha)$ be the corresponding constant when $|z_k|=1$. Equality of asymptotic constants conj…

Exact unit-circle asymptotic conjecture. One can choose $B(\alpha)=\underline B(\alpha)=\overline B(\alpha)$ in Theorem $avs$. This conjecture would close the gap between the known…

Exact asymptotic formula conjecture. One can choose $A(\alpha)=\underline A(\alpha)=\overline A(\alpha)$ in Theorem $u77$. This would determine the true asymptotic constant for eve…

Conjectured optimal bound. For $1/\sqrt e\leqslant \alpha\leqslant 1$, one has

Let $N,d,t,L$ be positive integers, let $\Psi=(\psi_1,\ldots,\psi_t)$ be a system of affine-linear forms with $\|\Psi\|_N\leqslant L$, and let $K\subset[-N,N]^d$ be a convex body.…

Let $s\geqslant 1$. An $s$-step nilmanifold is a quotient $G/\Gamma=(G/\Gamma,d_{G/\Gamma})$ with smooth metric, and an $s$-step nilsequence is a sequence of the form…