Limitations to equidistribution in arithmetic progressions

Abstract: 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.

classical analysis and ODEscombinatoricsnumber theory

Audience: researchers in the topic

Special Functions and Number Theory seminar

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