Pretentiously detecting power cancellation

TL;DR

This paper explores the concept of pretentiousness in multiplicative functions to better understand power cancellation, introducing new notions that improve detection across broader classes of functions.

Contribution

It develops two new notions of pretentiousness that enable more effective detection of power cancellation in multiplicative functions, expanding applicability.

Findings

New notions of pretentiousness detect power cancellation

Broader class of multiplicative functions analyzed

Enhanced understanding of cancellation behavior

Abstract

Granville and Soundararajan have recently introduced the notion of pretentiousness in the study of multiplicative functions of modulus bounded by 1, essentially the idea that two functions which are similar in a precise sense should exhibit similar behavior. It turns out, somewhat surprisingly, that this does not directly extend to detecting power cancellation - there are multiplicative functions which exhibit as much cancellation as possible in their partial sums that, modified slightly, give rise to functions which exhibit almost as little as possible. We develop two new notions of pretentiousness under which power cancellation can be detected, one of which applies to a much broader class of multiplicative functions.