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.

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Last updated: 03 Feb 2023