9 problems

Let $a<q$ be integers with $\gcd(a,q)=1$, and let $q\leq x$. Write $\psi(x,q,a)$ for the weighted prime-counting function in the progression $a\pmod q$, $\pi(x,q,a)$ for the corres…

Let $p,q,r$ be primes. The ternary progression Goldbach conjecture. Every odd integer $n>5$ can be written as … with … This is stated as a ternary analogue of one of the explicit m…

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…

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

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…

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…