6 problems

Let $R(n)$ denote the number of representations of an integer $n$ as a sum of two primes. Weak Hardy–Littlewood conjecture. There exists an absolute constant $c_3>0$ such that, for…

Define the weighted Goldbach representation function … and the singular series … where $C_2$ is the twin prime constant. Friedlander–Goldston–Iwaniec–Suriajaya conjecture. There ex…

Let $R(n)$ denote the number of representations of an integer $n$ as a sum of two primes, and let $\varphi$ denote Euler's totient function. Jia's strengthened Hardy–Littlewood con…

Let $g\in\mathbb{Z}_{\geq 1}$. For a CM-field $E$ with maximal totally real subfield $F$, write $\chi_{E/F}$ for the quadratic character associated to $E/F$ and…

Let $-d\in\mathbb{Z}_{\leq -2}$ be a fundamental discriminant, let $\chi_d$ be the quadratic character associated to $\mathbb{Q}(\sqrt{-d})/\mathbb{Q}$, and let $L(s,\chi_d)$ be it…

Finiteness conjecture for exceptional characters. For an exponent such as $100$, there are at most finitely many exceptional characters.