ИСТИНА

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−ϵ} 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−φ(n)=c (here G(k) is the number of ways to represent k as a sum of two primes) is G(c+1)+O(c^{3/4+o(1)}) To obtain this we use so-called multiplicative graphs.