3 problems

Let $g(n)$ denote the number of representations of an even integer $n$ as a sum of two primes. Exceptional weak Hardy--Littlewood conjecture. Suppose that $x$ is sufficiently large…

Let $g(n)$ denote the number of representations of an even integer $n$ as a sum of two primes. Fei's weakened Hardy--Littlewood conjecture. There exists a positive constant $delta$…

Let $L(s,chi)$ be a Dirichlet $L$-function, and call a zero on the critical line if its real part is $1/2$ and real if it lies on the real axis. Modified Generalised Riemann Hypoth…