Properties of the function f(x)=x/π(x)

Abstract

We obtain the asymptotic estimations for ∑k=2nf(k) and ∑k=2n1/f(k), where f(k)=k/π(k), k≥2. We study the expression 2f(x+y)−f(x)−f(y) for integers x,y≥2 and as an application we make several remarks in connection with the conjecture of Hardy and Littlewood.