On the difference between the number of prime divisors at consecutive integers

Abstract

Suppose thatg(n) is equal to the number of divisors ofn, counting multiplicity, or the number of divisors ofn, a≠0 is an integer, andN(x,b)=|{n∶n≤x, g(n+a)−g(n)=b orb+1}|. In the paper we prove that supbN(x,b)≤C(a)x)(log log 10x)−1/2 and that there exists a constantC(a,μ)>0 such that, given an integerb |b|≤μ(log logx)1/2,x≥xo, the inequalityN(x,b)≥C(a,μ)x(log logx(−1/2) is valid.