Multiplicative functions that are close to their mean

Abstract

We introduce a simple sieve-theoretic approach to studying partial sums of multiplicative functions which are close to their mean value. This enables us to obtain various new results as well as strengthen existing results with new proofs. As a first application, we show that for a completely multiplicative function $f:\mathbb{N}\to \{-1,1\},$ \begin{align*} \limsup_{x\to\infty}\Big|\sum_{n\leq x}μ^2(n)f(n)\Big|=\infty. \end{align*} This confirms a conjecture of Aymone concerning the discrepancy of square-free supported multiplicative functions. Secondly, we show that a completely multiplicative function $f:\mathbb{N}\to \mathbb{C}$ satisfies \begin{align*} \sum_{n\leq x}f(n)=cx+O(1) \end{align*} with $c\ne 0$ if and only if $f(p)=1$ for all but finitely many primes and $|f(p)|<1$ for the remaining primes. This answers a question of Ruzsa. For the case $c=0,$ we show, under the additional hypothesis $$\sum _p\frac{1-|f(p)|}{p}<\infty ,$$ that $f$ has bounded partial sums if and only if $f(p)=\chi (p)p^{it}$ for some non-principal Dirichlet character $\chi$ modulo $q$ and $t\in \mathbb{R}$ except on a finite set of primes that contains the primes dividing $q$, wherein $|f(p)|<1.$ This provides progress on another problem of Ruzsa and gives a new and simpler proof of a stronger form of Chudakov's conjecture. Along the way we obtain quantitative bounds for the discrepancy of the generalized characters improving on the previous work of Borwein, Choi and Coons.

Comments of the referee have been incorporated. Trans. of the AMS., to appear

References in corpus (2)

• A note on multiplicative functions resembling the Möbius function
• The Erdős discrepancy problem over the squarefree and cubefree integers

Cited by in corpus (4)

• On the Liouville function at polynomial arguments
• Expansion, divisibility and parity
• Furstenberg systems of pretentious and MRT multiplicative functions
• Partition regularity of Pythagorean pairs