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

Abstract

The following general problem is studied: Given a positive integer $c$ and two multiplicative functions $f$ and $g$, it is required to determine for what values of $n$ the equality $f(n)-g(n)=c$ holds. It is proved that, under certain constraints on the functions $f$ and $g$ and the solutions (in particular, under the constraint $f(n)>g(n)$ for $n>1$), this equation has at most $c^{1-\epsilon}$ solutions.

For the equation $n-\varphi(n)=c$, it is proved that the number of solutions equals

$$G(c+1)+O(c^{3/4+o(1)}),$$

where $G(k)$ is the number of ways in which $k$ can be represented as a sum of two primes.

This result is based on an assertion concerning configurations of points and straight lines.