20 problems

Let $d\geq1$ be an integer. For every prime power $q$, let $\Omega(q)\subset\mathbb Z/q\mathbb Z$ contain at most $d$ residues, and define $\Omega(b)$ for general $b$ by the Chines…

Anatomy-of-integers conjecture. For a fixed integer $s\geqslant 1$ and any integers $2\leqslant h<u$, we have

Let $\mathcal{L} = \{L_1,\ldots,L_K\}$ be a set of distinct linear forms $L_k(n) = a_kn+b_k$, where the coefficients $a_k,b_k$ are positive integers. For a prime $p$, let…

Let $\varepsilon\in(0,1)$, and let $v=\log x/\log z$ be fixed. Write $\Psi(x',z;a,q)$ for the count of $z$-friable integers up to $x'$ in the reduced residue class $a\pmod q$, and…

Let $p\ge 3$ be prime and set $N_p=2^p-1$. Write $\omega(N_p)$ for the number of distinct prime factors of $N_p$. The distinct-factor bound conjecture. There exists $p_0$ such that…

Cyclotomic prime-divisor sparsity conjecture. One has

Extended tail-bound conjecture. As $x\to\infty$,

Conjectural tail estimates. For every $\epsilon>0$,

Let $(\varphi_k)_k$, $(Q_k)_k$, and $(f_k)_k$ satisfy the hypotheses of the quadratic Hecke-sum conjecture: the forms $\varphi_k$ are non-dihedral, the $Q_k$ are irreducible intege…

Let $m,d$ be natural numbers with $d\geqslant m+2$, and let $L:\mathbb{R}^d\to\mathbb{R}^m$ be a surjective linear map. Let $V^*_{\operatorname{degen}}(m,d)$ denote the degenerate…

Constancy conjecture for $\Omega_k$ and $\Omega_k^{\mathrm{ext}}$. The function $\Omega_k(\vartheta,\vartheta_0)$ is constant with respect to $\vartheta_0$. The same applies to…

Let $G\subset GL_n(\mathbb{R})$ be the group of real points of an algebraically connected, algebraically simply connected, absolutely almost simple linear algebraic group defined o…

Let $f\in\mathbb{Z}[x_1,\ldots,x_h]$ be primitive and squarefree, and let $\mathbb{Z}^h(X)=\{{\rm v}\in\mathbb{Z}^h:|{\rm v}|\leq X\}$. Squarefree conjecture, alternative version.…

Let $h\in\{1,2\}$, let $f\in\mathbb{Z}[x_1,\ldots,x_h]$ be primitive and squarefree, and let $\mathbb{Z}^h(X)=\{{\rm v}\in\mathbb{Z}^h:|{\rm v}|\leq X\}$. Define…

Let $0<\varpi<1/4$, and let $\delta>0$ be a sufficiently small fixed quantity depending on $\vartheta$. Let $x$ tend to infinity, let $P$ be a square-free number all of whose prime…

Let $\pi(x)$ be the prime-counting function. For a positive integer $c$, let $I_k=[(k-1)c,kc)$ and define $\chi_{I_k}=1$ if $I_k$ contains one or more primes, and $\chi_{I_k}=0$ ot…

Let $p_n$ denote the $n$th prime, and let $\mathcal{P}_{p_n}$ denote the corresponding sieve pattern. Modal multiplicity conjecture. The number $6$ occurs most frequently among the…

Generalized Chowla's conjecture. For fixed $0\le\kappa_1<\kappa_2$ and $y=\exp((\log x)^\delta)$ with fixed $0<\delta<1$,

Let ${\mathcal G}(p)$ denote the cycle of gaps at a sieve stage, let $s$ be a constellation in ${\mathcal G}(p)$, and let $\sigma(s)$ be the sum of its gaps. Assume $\sigma(s)<2p$.…

Growth conjecture for $\mathcal{P}_r$. For each $r$, there is a number $\delta_r>0$ such that