Multiplicative functions resembling the M\"{o}bius funciton

TL;DR

This paper investigates multiplicative functions similar to the Möbius function, establishing bounds on their summatory functions that show significant cancellation, with results depending on the Riemann Hypothesis.

Contribution

It provides new upper and lower bounds for the summatory functions of these functions, extending understanding of their oscillatory behavior compared to the classical Möbius function.

Findings

Summatory function is O(x^{1/3+ε}) under RH

Unconditionally, the summatory function is Ω(x^{1/4})

Demonstrates better cancellation than square-root saving

Abstract

A multiplicative function $f$ is said to be resembling the M\"{o}bius function if $f$ is supported on the square-free integers, and $f(p)=\pm 1$ for each prime $p$. We prove $O$- and $\Omega$-results for the summatory function $\sum_{n\leq x} f(n)$ for a class of these $f$ studied by Aymone, and the point is that these $O$-results demonstrate cancellations better than the square-root saving. It is proved in particular that the summatory function is $O(x^{1/3+\varepsilon})$ under the Riemann Hypothesis. On the other hand it is proved to be $\Omega(x^{1/4})$ unconditionally. It is interesting to compare these with the corresponding results for the M\"{o}bius function.