New insight into results of Ostrowksi and Lang on sums of remainders using Farey sequences

Abstract

The sums $S(x,t)$ of the centered remainders $kt–\lfloor kt\rfloor –1/2$ over $k\le x$ and corresponding Dirichlet series were studied by A. Ostrowski, E. Hecke, H. Behnke, and S. Lang for fixed real irrational numbers $t$. Their work was originally inspired by Weyl’s equidistribution results modulo 1 for sequences in number theory.

In a series of former papers, we obtained limit functions which describe scaling properties of the Farey sequence of order $n$ for $n\to \mathrm{\infty }$ in the vicinity of any fixed fraction $a/b$ and which are independent of $a/b$. We extend this theory on the sums $S(x,t)$ and also obtain a scaling behavior with a new limit function. This method leads to a refinement of results given by Ostrowski and Lang and establishes a new proof for the analytic continuation of related Dirichlet series. We will also present explicit relations to the theory of Farey sequences.