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 with a "level of distribution" 1/2. In 1989, building on an idea of Maier, Friedlander and Granville showed that such equidistribution results fail if the range of the moduli q is extended up to x/(log x)^B for any B>1.  We discuss variants of this result and applications to specific functions. This is joint work with Aditi Savalia.