Primes in arithmetic progressions to large moduli and refinements of Harman's sieve

We study the average distribution of primes of size $x$ in arithmetic progressions to moduli larger than $x^{\frac{1}{2}}$. Using arithmetic information from the works of many authors together with different variants of the original Harman’s sieve, we construct suitable majorants and minorants for the prime indicator function ${\mathbb{1}}_p(n)$ that satisfy Bombieri–Vinogradov type mean value theorems with different types of moduli. Specifically, we obtain some mean value theorems for primes with bilinear forms of moduli up to $x^{\frac{9}{17}}$ or with trilinear forms of moduli up to $x^{\frac{17}{32}}$. As a by-product, we obtain new upper and lower bounds for $\pi (x;q,a)$ that hold for almost all moduli $q$.