The average number of divisors of the Euler function

TL;DR

This paper investigates the average number of divisors of the Euler Phi function and Carmichael Lambda function, improving lower bounds and providing heuristic evidence that the existing upper bounds are accurate.

Contribution

It improves the lower bounds and offers heuristic support for the accuracy of existing upper bounds on the average number of divisors of these functions.

Findings

Improved lower bounds for the average number of divisors.

Heuristic argument supporting the tightness of existing upper bounds.

Enhanced understanding of divisor distribution for Euler and Carmichael functions.

Abstract

The upper bound and the lower bound of average numbers of divisors of Euler Phi function and Carmichael Lambda function are obtained by Luca and Pomerance (see \cite{LP}). We improve the lower bound and provide a heuristic argument which suggests that the upper bound given by \cite{LP} is indeed close to the truth.