The exceptional weak Hardy--Littlewood conjecture

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 and $q\leq x/4$. Then, with at most $x/(8q)$ exceptions, one has

$$g(n)\gg \frac{n}{\log^2 n}$$

for the multiples $n$ of $q$ in the interval $x/2<n\leq x$. This hypothesis weakens the preceding weak Hardy--Littlewood conjecture by allowing a controlled exceptional set and is used in the paper to extend Fei's conditional Siegel-zero bound to suitable composite moduli.