Farey fractions with power denominator and Waring’s problem

Wednesday, 26 July 2023

Abstract

A Farey fraction of order q is a reduced fraction between 0 and 1 with denominator q. We are particularly interested in how the set of Farey fractions of order at most Q is distributed along the unit interval. Indeed, Franel and Landau each gave equivalent statements for RH in terms of the distribution of Farey fractions. While this is currently out of reach, we already know that “RH for arithmetic progressions is true on average” (the Bombieri-Vinogradov theorem). A key input for this averaged version of RH is a trivial fact about the distribution of Farey fractions. However, if we restrict the moduli of the arithmetic progressions over which we average, then we need to understand the distribution of Farey fractions with restricted denominators. Such a restrictions transforms the once trivial input into a challenging one.

To discuss my recent joint work with Stephan Baier, we will focus on the distribution of Farey fractions whose denominators are kth powers. After briefly describing Waring’s problem, we will form the connection between the two problems. Then we just have to Cauchy-Schwarz a couple of times and apply known estimates. The efficiency of our method will be stressed in the case of squares (k=2).