34 problems

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

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…

Goldbach representation conjecture. The bound

Weighted Elliott–Halberstam conjecture. For every $A>0$, there is $B=B(A)>0$ such that all three sums

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

Sun's conjecture. Every integer $n>1$ has a decomposition $n=x+y$ with integers $x,y>0$ such that $x+ny$ and $x^2+ny^2$ are simultaneously prime.

Backwards-coprime proportion conjecture.

Let $a>1$ and let $b$ be a non-zero integer. Hooley considered the sequence $2^n+5$ as $n$ runs over the positive integers. Hooley's conjecture. Almost all numbers in the sequence…

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 $x\to\infty$,

Let $\mathcal{H}=\{h_1,\ldots,h_k\}$ be a set of distinct integers. It is admissible if $\#\{\mathcal{H}\,\, (\operatorname{mod}\,p)\}<p$ for every prime $p$. The prime $k$-tuples…

Prime-sieve conjecture. The sufficient condition for the existence of such an element $\alpha$ is

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…

Quantitative twin-prime lower-bound conjecture. For each integer $a>0$ there exists an integer $N_2(a)$ such that

Twin prime conjecture. There exist infinitely many twin primes.

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…

Let $L$ be a surjective linear map as in Theorem 1, and let $w$ be the auxiliary function occurring there. Write $V^*_{\operatorname{degen}}(m,d)$ for the degenerate locus of linea…

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 $\mathcal{N}$ be a set of $Z$ integers contained in $[M+1,M+N]$, and let $\omega(p)$ denote the number of residue classes modulo $p$ containing no element of $\mathcal{N}$. Def…

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…

For $k\geq2$, let ${\mathcal R}_k$ be the domain used to define $M_k$, and let a positive eigenfunction mean a function $F:{\mathcal R}_k\to(0,+\infty)$, extended by zero outside…