21 problems

Let $X$ be large, let $\delta$ be a positive real parameter, and let $\psi(x)=\sum_{n\leq x}\Lambda(n)$ be Chebyshev's function. For fixed $\epsilon>0$ and $\lambda>0$, define…

Let $X$ be large, let $h$ be a positive real parameter, and let $\psi(x)=\sum_{n\leq x}\Lambda(n)$ be Chebyshev's function. For fixed $\epsilon>0$ and $\lambda>0$, define…

Let $A_N(\gamma)$ be the oscillatory phase average associated with the centred layer moment, let $X$ be the relevant scale, and let $R$ be the corresponding correlation quantity. S…

Let $F_N(r)$ denote the empirical distribution of $q_{\min}\bmod \ell$, let $\pi_{\mathrm{perm}}(r)$ be the permitted-residue distribution, and let $C_{\mathrm{opt}}(k,\ell)$ be a…

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 $W\geq 1$ be fixed, and let $b_1,b_2$ be fixed distinct integers coprime to $W$. For each prime $p$, let $\nu_p(W\cdot+b_1,W\cdot+b_2)$ be the number of distinct residue classe…

Let $\Lambda(n)$ denote the von Mangoldt function. For $\delta\in(0,1)$, set $H=X^\delta$. Goldston–Montgomery conjecture. As $X\to\infty$, … This predicts the variance of sums of…

Let $E/{\mathbb Q}$ be an elliptic curve, and for each prime $p$ of good reduction let $\widetilde{E}({\mathbb F}_p)$ denote the group of rational points of its reduction modulo…

Let $L$, $\mathbf{v}$, $F$, and $G$ be as in Theorem 1, without assuming that $L$ has algebraic coefficients. Let $W=\prod_{p\leq w}p$. Transcendental-case conjecture. There should…

Let $k$ be a positive integer and let $a$ be an integer coprime to $k$. The primes $p_1$ and $p_2$ are required to satisfy $p_1,p_2\leq k$. Erdős–Odlyzko–Sárközy conjecture. For al…

Let $x$ be real, let $q<x$ be a positive integer, let $a$ satisfy $(a,q)=1$, and let $\pi(x;q,a)$ count primes $p\leq x$ with $p\equiv a\pmod q$. Let $\varphi(q)$ denote Euler's to…

Let $N$ be large, and let $[a,b]$ be an interval contained in $[N,2N]$ whose length is at most $N^{1/2+c}$ for some absolute constant $c>0$. Square-root barrier conjecture. There e…

Let $a_1, \ldots, a_t$ be the longest sequence for which … Probably $t = \pi(n)$. Can one even prove $t < (1 + o(1))\pi(n)$ or at least $t = o(n)$? This latest conjecture will prob…

A purely computational problem (this problem cannot be attacked by other means at present). Call a prime $p$ good if every even number $2r \leq p - 3$ can be written in the form…

Perhaps the following rather silly conjecture could be added. Is it true that the set of odd integers not of the form $2^k + p$ is the not necessarily disjoint union of an infinite…

I just discovered in it a forgotten conjecture of mine, which might still be of interest. Let $\{p_n\}$ be the sequence of consecutive primes. Is it true that…

One could ask the following (probably unattackable) problem. Is it true that there is an $r$ so that every integer is the sum of a prime and $r$ or fewer powers of 2.

Crocker [16] proved that there are infinitely many odd integers not of the form $2^{k_1} + 2^{k_2} + p$, but his proof only gives that the number of integers $\leq x$ not of the fo…

Let $2 = p_1 < p_2 < \cdots$ be the sequence of consecutive primes. [$p_{n+1} - p_n = d_n$.] Turán and I [15] proved that the inequalities $d_{n+1} > d_n$ and $d_{n+1} < d_n$ both…

Let $2 = p_1 < p_2 < \cdots$ be the sequence of consecutive primes. [...] $p_{n+1} - p_n = d_n$. [...] It seems likely that $d_n/\log n$ is everywhere dense in $(0, \infty)$. Ricci…

For every $c>0$, there are infinitely many indices $n$ such that …