On Veech's proof of Sarnak's theorem on the M\"{o}bius flow

TL;DR

This paper explains Veech's proof of Sarnak's theorem on the M"obius flow, showing the uniqueness of the measure and connecting Sarnak's and Chowla's conjectures through Tao's logarithmic theorem, with implications for the M"obius function's behavior.

Contribution

It provides a detailed presentation of Veech's proof and establishes the equivalence between Sarnak's and Chowla's conjectures via Tao's logarithmic theorem.

Findings

Uniqueness of the admissible measure on the M"obius flow.

Equivalence of Sarnak's and Chowla's conjectures using Tao's theorem.

Existence of a subsequence where Chowla's conjecture holds if the even logarithmic Sarnak conjecture is true.

Abstract

We present Veech's proof of Sarnak's theorem on the M\"{o}bius flow which say that there is a unique admissible measure on the M\"{o}bius flow. As a consequence, we obtain that Sarnak's conjecture is equivalent to Chowla conjecture with the help of Tao's logarithmic Theorem which assert that the logarithmic Sarnak conjecture is equivalent to logaritmic Chowla conjecture, furthermore, if the even logarithmic Sarnak's conjecture is true then there is a subsequence with logarithmic density one along which Chowla conjecture holds, that is, the M\"{o}bius function is quasi-generic.