17 problems

Uniform Hardy–Littlewood prime $k$-tuple conjecture. There exists an absolute constant $0<\delta<1/2$ such that

Exceptional-set growth conjecture. As $m\to\infty$,

Friedlander–Goldston conjecture. The Hooley asymptotic $G(X,Q)\sim X\log Q$ should hold when $X^{1/2+\epsilon}<Q<X$, and, when $X^{1/2+\epsilon}<Q<X^{1-\epsilon}$,

Let $r\geq 3$, let $k$ be a modulus, and let $a_1,\ldots,a_r$ be reduced residue classes modulo $k$. Write $\{k;a_1,\ldots,a_r\}$ for the corresponding prime-number race, and let…

Zhang's least-prime bound conjecture. For every sufficiently large positive integer $k$, namely $k>c_1$ for a positive constant $c_1$, one has

Let ${\cal P}_2$ denote the set of primes having $2$ as a primitive root, and let $A$ be the constant defining the density of ${\cal P}_2$. Rodier's conjecture. The density of the…

Weak pair correlation for Dirichlet $L$-functions. Under GRH, as $x\to\infty$,

Chowla's conjecture. For every $\varepsilon>0$, $L=1+\varepsilon$ is a Linnik constant.

Let $q$ be a positive integer and let $b$ be coprime to $q$. Define $L(q,b)$ to be the least prime congruent to $b$ modulo $q$. Least-prime-in-progression conjecture. One should ha…

Let $E_{a,b,m}$ be the exceptional set of positive even integers in the residue class $a+b\pmod m$ that do not have a representation $n=p+q$ with primes $p\equiv a\pmod m$ and…

Let $p$ and $q$ be primes. The selected-moduli refined Goldbach conjecture. The following assertions hold: 1. Every even $n\equiv0\pmod3$ except $n=6$ has a representation with…

Let $p$ and $q$ be primes. The mod-$4$ refined Goldbach conjecture. The following assertions hold: 1. Every even $n>4$ has a representation $n=p+q$ with $p\equiv3\pmod4$. 2. Every…

Suppose that $\alpha(m)$ and $\beta(n)$ satisfy the hypotheses stated earlier in the source, and write $(\alpha*\beta)(n)$ for their convolution. The Bombieri–Friedlander–Iwaniec c…

Let $r\geq 3$ and let $a_1,\ldots,a_r$ be distinct nonzero integers. Define $\mathcal{Q}_{a_1,\ldots,a_r}$ to be the set of positive integers $q$ such that the $a_i$ are distinct m…

Let $k$ be a modulus, let $\varphi(k)$ denote Euler's totient, and let $a_1,\ldots,a_{\varphi(k)}$ be a permutation of the reduced residue classes modulo $k$. Write $\pi(m,k,a)$ fo…

Let $E(x;q,a)$ be the discrepancy … where $\pi_{q,a}(x)$ counts primes at most $x$ congruent to $a$ modulo the prime $q$. Mean-square comparison conjecture. There exists an…

Let $q$ be a positive integer and let $a$ be coprime to $q$. Define … where $\Lambda$ is the von Mangoldt function. For $\theta\in[1/2,1)$, say that the primes have level of distri…