On Differences of Multiplicative Functions and Solutions of the Equation $n-\varphi(n) = c$

Abstract

We will study the solutions to the equation $f(n) - g(n) = c$, where $f$ and $g$ are multiplicative functions and $c$ is a constant. More precisely, we prove that the number of solutions does not exceed $c^{1-\epsilon}$ when $f, g$ and solutions $n$ satisfy some certain constraints, such as $f(n) > g(n)$ for $n > 1$. In particular, we will prove the following estimate: the number of solutions to the equation $n - \varphi(n) = c$ is: $$ G(c + 1) + O(c^{0.75 + o(1)}), $$ where $G(k)$ is the number of ways to represent $k$ as a sum of two primes. This result is based on some properties of configurations of points and lines.