Distribution of rational numbers in short intervals

Abstract

A simplified proof for a well-distribution property for rational numbers is given and a connection with Riemann’s Hypothesis is pointed out. More precisely, we consider rational numbers with denominators of a given order of magnitude and show that the number of such numbers lying in a short interval of given length is normally close to its expectation in a mean square sense. The proof is elementary, using only Fourier series and Ramanujan sums. At the end of the paper, a variant of the circle method is discussed as an application.