15 problems

Let $p_n$ denote the $n$th prime, and let $\varepsilon>0$. Poisson Tail Conjecture. For $1\leq H\leq(\log X)^{2-\varepsilon}$, the two gap-counting quantities satisfy … … For…

Let $K$ be a totally real number field of degree $n$. For an odd prime $p$ unramified in $K/\mathbb{Q}$ and a place $\mathfrak{p}$ above $p$, let $\tau^{\mathfrak p}$ be the map ap…

Let $f:\mathbb{N}\to\mathbb{N}_0$ satisfy $0\leqslant f(n)\leqslant n$, let $\Omega_f$ be the associated random space of numerator sets, and let $W^P(\Psi)$ denote the set of numbe…

For primes $p\ne\ell$, let $D_p=\dim_{\mathbb{Q}}S_2(\Gamma_0(p);\mathbb{Q})$, let $D_p^\pm=\dim_{\mathbb{Q}}S_2(\Gamma_0(p);\mathbb{Q})^\pm$, and define … where the tilded polynom…

Let $C(n)$ range over the Apéry-like sequences listed in Tables 2 and 3 of the source, including $(\eta)$ and the sequences $(\delta)$, $(\alpha)$, $(\epsilon)$, $(\zeta)$, and…

Let $A(n)$ denote the Apéry numbers, and let a prime divide the sequence if it divides at least one term $A(n)$. The Apéry-number proportion conjecture. The proportion of primes th…

Random-multiplicative-function moment conjecture. As $N\rightarrow\infty$,

Let $\Sigma=(H,K_0,\sigma)$ be a situation satisfying the stated character-theoretic assumptions, let ${\mathcal{O}}={\mathcal{O}}(\Sigma)$, and let ${\mathfrak{p}}$ be a prime ide…

For each prime $p$, let ${\mathcal C}_p$ be the set defined in the paper that records the relevant sporadic roots or residue classes. Average-size conjecture for ${\mathcal C}_p$.…

For each prime $p$, let $f_p$ be the polynomial defined in the paper, and let $\rho(m)$ be the relative density of primes $p$ for which $f_p$ has exactly $m$ roots; equivalently, t…

Let $q$ be a prime, let $\mathbb F_q$ be the finite field with $q$ elements, let $d=\pi(q-1)$, and let $L_n$ be the linear forms from the paper on $\mathbb F_q^d$. For…

3x+1 maximum excursion constant conjecture. The maximum excursion constant $\rho$ is finite and

Let $B_n$ be the maximum at generation $n$ in the branching random walk considered in the paper, and let $b_n$ be its deterministic centering term. Branching-random-walk convergenc…

For a prime $p$, let $H(p)$ be the height of its Pratt tree, let $\log_2 p=\log\log p$ and $\log_3 p=\log\log\log p$, and write $\pi(x)$ for the number of primes at most $x$. Refin…

Let $a_1,a_2,\ldots$ be independent uniform random integers in $\{1,\ldots,x\}$, and let $T$ be the least $t$ such that some subsequence of $a_1,\ldots,a_t$ has product equal to a…