Complex valued multiplicative functions with bounded partial sums

TL;DR

This paper constructs a class of multiplicative functions with unbounded modulus that have bounded partial sums, are periodic, and exhibit specific zero-sum properties, extending understanding of their behavior beyond traditional bounded cases.

Contribution

It introduces a new class of multiplicative functions with unbounded modulus that are periodic and have bounded partial sums, a novel extension in the study of such functions.

Findings

Functions are periodic with zero sum in each period.

Partial sums have an omega lower bound.

Upper bounds relate to the Dirichlet divisor problem error term.

Abstract

We present a class of multiplicative functions $f:\mathbb{N}\to\mathbb{C}$ with bounded partial sums. The novelty here is that our functions do not need to have modulus bounded by $1$. The key feature is that they pretend to be the constant function $1$ and that for some prime $q$, $\sum_{k=0}^\infty \frac{f(q^k)}{q^k}=0$. These combined with other conditions guarantee that these functions are periodic and have sum equal to zero inside each period. Further, we study the class of multiplicative functions $f=f_1\ast f_2$, where each $f_j$ is multiplicative and periodic with bounded partial sums. We show an omega bound for the partial sums $\sum_{n\leq x}f(n)$ and an upper bound that is related with the error term in the classical Dirichlet divisor problem.