Mean value theorems for binary Egyptian fractions II

TL;DR

This paper investigates the distribution of solutions to a specific binary Egyptian fraction equation, establishing mean value theorems and showing that the solution count behaves similarly to the divisor function, with implications for understanding its probabilistic distribution.

Contribution

It proves mean value theorems for the second moment of the solution count and demonstrates a Gaussian distribution for its logarithm, extending classical results to Egyptian fractions.

Findings

Log R(n;a) follows a Gaussian distribution with specified mean and variance.

The behavior of R(n;a) resembles the divisor function d(n^2).

Established mean value theorems for the second moment of R(n;a).

Abstract

In this article, we continue with our investigation of the Diophantine equation $\frac{a}n=\frac1x+\frac1y$ and in particular its number of solutions $R(n;a)$ for fixed $a$. We prove a couple of mean value theorems for the second moment $(R(n;a))^2$ and from which we deduce $\log R(n;a)$ satisfies a certain Gaussian distribution with mean $\log 3\log\log n$ and variance $(log 3)^2\log\log n$, which is an analog of the classical theorem of Erd\H os and Kac. And finally these results in all suggest that the behavior of $R(n;a)$ resembles the divisor function $d(n^2)$ in various aspects.