Farey Statistics in Time n^{2/3} and Counting Primitive Lattice Points in Polygons

TL;DR

This paper introduces efficient algorithms for computing Farey sequence statistics in time O(n^{2/3}) and explores counting primitive lattice points in polygons, advancing computational number theory methods.

Contribution

It improves the computational complexity for Farey sequence statistics and initiates the study of counting primitive lattice points in polygons.

Findings

Farey sequence algorithms run in O(n^{2/3}) time.

Enhanced algorithms outperform previous O(n^{3/4}) methods.

New approach to counting primitive lattice points in planar shapes.

Abstract

We present algorithms for computing ranks and order statistics in the Farey sequence, taking time O (n^{2/3}). This improves on the recent algorithms of Pawlewicz [European Symp. Alg. 2007], running in time O (n^{3/4}). We also initiate the study of a more general algorithmic problem: counting primitive lattice points in planar shapes.