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

## TL;DR

This paper refines and extends previous results on sums of centered remainders related to Farey sequences, providing new limit functions, a new proof of Dirichlet series continuation, and explicit connections to Farey sequence theory.

## Contribution

It introduces a new limit function describing the scaling behavior of sums of remainders and offers a refined analysis and proof of Dirichlet series properties related to Farey sequences.

## Key findings

- New limit function for sums of remainders
- Refined results on Farey sequence scaling
- Explicit relations to Farey sequence theory

## Abstract

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 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.