We obtain the asymptotic estimations for ∑ k = 2 n f ( k ) and ∑ k = 2 n 1 / f ( k ) , where f ( k ) = k / π ( k ) , k ≥ 2 . We study the expression 2 f ( 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.