A Mean Value Theorem for the Diophantine Equation $axy-x-y=n$

TL;DR

This paper derives an asymptotic formula for the average number of solutions to a specific Diophantine equation involving fixed and varying parameters, advancing understanding of its solution distribution.

Contribution

The paper introduces a new asymptotic formula for the average solutions of the equation $axy - x - y = n$, providing a novel analytical approach.

Findings

Established an asymptotic estimate for the number of solutions

Analyzed the solution behavior as n varies

Enhanced understanding of the equation's solution distribution

Abstract

In this paper, we prove an asymptotic formula for the average number of solutions to the Diophantine equation $axy-x-y=n$ in which $a$ is fixed and and $n$ varies.