Properties of the function \(f(x) = x/\pi(x)\) (Q1607744)

In this paper the author proves asymptotic expansions for \(\sum_{k=2}^n f(k)\) and \(\sum_{k=2}^n 1/f(k)\), where \(f(k) = k/\pi(k)\), \(k \geq 2\). He studies the expression \(2f(x+y)-f(x)-f(y)\) for integers \(x,y \geq 2\) and as an application gives several remarks in connection with the Hardy-Littlewood conjecture.