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

The sums \( S(x, t) \) of the centered remainders \( kt – \lfloor kt \rfloor – 1/2 \) over \( k \leq x \) 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 \( n \to \infty \) 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.