A note on correlations of arithmetic functions

TL;DR

This paper explores weight functions with transitional correlation behavior between weak and strong with the Liouville function, and introduces a binary problem bridging Chowla's conjecture and the twin prime conjecture, informed by recent parity-breaking results.

Contribution

It introduces new weight functions exhibiting transitional correlation behavior and proposes a binary problem connecting two major conjectures in number theory.

Findings

Identification of weight functions with transitional correlation properties

A binary problem interpolating between Chowla's conjecture and the twin prime conjecture

Insights based on recent parity-breaking results

Abstract

In this note we describe weight functions that exhibit a transitional behavior between weak and strong correlation with the Liouville function. We also describe a binary problem which may be considered as an interpolation between Chowla's conjecture for two-point correlations of the M\"obius function and the twin prime conjecture, in view of recent parity breaking results of K. Matom\"aki, M. Radziwi{\l}{\l} and T. Tao.