Number Theory Seminar: Distribution of Farey fractions

Seminar | April 10 | 3:40-4:40 p.m. | 748 Evans Hall

 Jeffrey C. Lagarias, University of Michigan

 Department of Mathematics

The Farey fractions of level $n$ are the set of rationals in $[0,1]$ in lowest terms having denominator at most $n$. It is known that a measure of equally weighted point masses (of total mass 1) on the points of the Farey sequence $F_n$ converges to the uniform distribution on $[0,1]$ as $n$ goes to infinity. The Riemann hypothesis is equivalent to suitably fast rates of convergence (to zero) of certain statistics measuring distance to uniform distribution, given by theorems of Franel (1924) and Landau (1924) . This talk addresses a toy model consisting of unreduced Farey fractions (allowing fractions not in lowest terms) and studies similar statistics.