Disjointness of the M\"obius Transformation and M\"obius Function

TL;DR

This paper investigates the distribution of sequences generated by Möbius transformations over finite fields, providing estimates that show these sequences are statistically independent from the Möbius function, supporting conjectures about their disjointness.

Contribution

It offers new nontrivial bounds on exponential sums for Möbius transformation sequences, advancing understanding of their independence from the Möbius function.

Findings

Sequences are disjoint from the Möbius function.

Provides bounds on exponential sums involving these sequences.

Supports conjecture of P. Sarnak on disjointness.

Abstract

We study the distribution of the sequence of elements of the discrete dynamical system generated by the M\"obius transformation $x \mapsto (ax + b)/(cx + d)$ over a finite field of $p$ elements. Motivated by a recent conjecture of P. Sarnak, we obtain nontrivial estimates of exponential sums with such sequences that imply that trajectories of this dynamical system are disjoined with the M\"obius function.