THE RAMANUJAN JOURNAL, Vol. 14, No. 2 (2007), 321-327.

Distribution of rational numbers in short intervals

MATTI JUTILA
Department of Mathematics, University of Turku, 20014 Turku, Finland
Email: jutila@utu.fi

Received February 3, 2004; Accepted July 15, 2004

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.

Keywords: Distribution of rational numbers, circle method

2000 Mathematics Classification: Primary 11K38; Secondary 11P55