THE RAMANUJAN JOURNAL, Vol. 14, No. 2 (2007),
Distribution of rational numbers in short intervals
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
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Updated Sat Sep 22 15:03:03 EDT 2007.