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

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 $n-\phi (n)=c$ 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.

7 pages