Explicit sieve estimates and nonexistence of odd multiperfect numbers of a certain form

TL;DR

This paper develops explicit asymptotic formulas for sieve functions and proves that odd multiperfect numbers of a specific form with certain properties do not exist.

Contribution

It introduces explicit sieve estimates and establishes the nonexistence of a class of odd multiperfect numbers with cubefree squared parts.

Findings

Explicit asymptotic formulas for sieve-related functions

Proof of nonexistence of certain odd multiperfect numbers

Advancement in understanding the structure of multiperfect numbers

Abstract

We prove explicit asymptotic formulae for some functions used in sieve methods and show that there exists no odd multiperfect number of abundancy four whose squared part is cubefree.