New insight into results of Ostrowski 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 $n\to \infty$ 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 behaviour 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.

Research article, 22 pages, 4 figures. v2: Final version as in "Online Journal of Analytic Combinatorics, Issue 15 (2020), #11"

References in corpus (1)

• On the meromorphic continuation of Beatty Zeta-Functions and Sturmian Dirichlet series