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 $q \le x^{1/2 -\epsilon }$
for any $\epsilon>0$
. 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 to $q \le x/ (\log x)^B$
for any $B>1$
. We discuss variants of this result and give some applications. This is joint work with Aditi Savalia.