Correlations of multiplicative functions and applications

TL;DR

This paper establishes asymptotic formulas for correlations of bounded pretentious multiplicative functions, leading to new characterizations, conjecture resolutions, and applications in additive number theory.

Contribution

It provides a novel asymptotic analysis of multiplicative function correlations and resolves longstanding conjectures by Erdős and Katai.

Findings

Characterization of multiplicative functions with bounded partial sums.

Resolution of Katai's conjecture on the behavior of multiplicative functions.

New proof of additive representation results without circle method.

Abstract

We give an asymptotic formula for correlations \[ \sum_{n\le x}f_1(P_1(n))f_2(P_2(n))\cdot \dots \cdot f_m(P_m(n))\] where $f\dots,f_m$ are bounded "pretentious" multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences:\ First, we characterize all multiplicative functions $f:\mathbb{N}\to\{-1,+1\}$ with bounded partial sums. This answers a question of Erd\H{o}s from $1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either $f(n)=n^s$ for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of K\'atai. Third, we apply our theorem to count the number of representations of $n=a+b$ where $a,b$ belong to some multiplicative subsets of $\mathbb{N}.$ This gives a new "circle method-free" proof of the result of Br\"udern.