On positive sequences of reals whose block sequence has an asymptotic distribution function

In this paper we study the properties of the unbounded sequence 0 < y₁ ≤ y₂ ≤ y₃ ≤ ⋯ of positive reals having asymptotic distribution function of the form x^λ. As a consequence, we immediately get information on the asymptotic behavior of the power means of order $r>0$ of function values of some arithmetic functions, e.g., the first n prime numbers or the values of the prime counting function.