On variants of Chowla's conjecture

TL;DR

This paper investigates shifted convolution sums of multiplicative functions with values in  and provides combinatorial proofs for recent results related to Chowla's conjecture, also analyzing their spectral properties.

Contribution

It offers new combinatorial proofs for recent findings on Chowla's conjecture and characterizes the spectrum of related convolution sums.

Findings

Provided combinatorial proofs of recent results on Chowla's conjecture

Determined the spectrum of shifted convolution sums for multiplicative functions

Enhanced understanding of the structure of multiplicative functions in this context

Abstract

We study the shifted convolution sums associated to completely multiplicative functions taking values in $\{\pm 1\}$ and give combinatorical proofs of two recent results in the direction of Chowla's conjecture. We also determine the corresponding "spectrum".