On multiplicative functions which are small on average

TL;DR

This paper investigates the behavior of completely multiplicative functions bounded within the unit disc, demonstrating that if their partial sums are small, then they are either small on average at primes or closely resemble a specific multiplicative function involving the Möbius function and a power of n.

Contribution

The paper establishes a dichotomy for such functions, showing they are either small on average at primes or pretentious to a specific form, extending understanding of multiplicative function behavior.

Findings

If the partial sums are small, then f(p) is small on average.

Alternatively, f pretends to be μ(n)n^{it} for some t.

Provides conditions under which multiplicative functions are either small or pretentious.

Abstract

Let $f$ be a completely multiplicative function that assumes values inside the unit disc. We show that if $\sum_{n<x} f(n) \ll x/(\log x)^A$, $x>2$, for some $A>2$, then either $f(p)$ is small on average or $f$ pretends to be $\mu(n)n^{it}$ for some $t$.