Mean Value Theorems for Binary Egyptian Fractions

TL;DR

This paper proves two mean value theorems related to the number of solutions of a specific Egyptian fraction Diophantine equation, enhancing understanding of their average behavior when parameters vary.

Contribution

It introduces new mean value theorems for solutions of the binary Egyptian fraction equation with varying parameters, expanding theoretical knowledge in this area.

Findings

Established mean value theorems for fixed and varying parameters

Provided asymptotic formulas for the average number of solutions

Enhanced understanding of the distribution of solutions in Egyptian fractions

Abstract

In this paper, we establish two mean value theorems for the number of solutions of the Diophantine equation $\frac{a}{n}=\frac{1}{x}+\frac{1}{y}$, in the case when $a$ is fixed and $n$ varies and in the case when both $a$ and $n$ vary.