Limitations to equidistribution in arithmetic progressions

It is well known that the prime numbers are equidistributed in arithmetic progressions. Such a phenomenon is also observed more generally for a class of arithmetic functions. A key result in this context is the Bombieri--Vinogradov theorem which establishes that the primes are equidistributed in arithmetic progressions ``on average" for moduli $q$ in the range $q\leq x^{1/2-\epsilon}$ for any $\epsilon > 0$. Building on an idea of Maier, Friedlander--Granville showed that such equidistribution results fail if the range of the moduli $q$ is extended to $q\leq x/(\log x)^B$ for any $B>1$. We discuss variants of this result and give some applications. This is joint work with my supervisor Akshaa Vatwani