The asymptotic form of the sum Σni=0i pn+ii two proofs - Kirschenhofer, Peter,; Larcombe, Peter J.,; Fennessey, Eric J., (Utilitas Mathematica, 2014)
The asymptotic form of the sum Σni=0i p (n+ii) is established in two quite different ways-by means of the longstanding Euler-Maclaurin summation formula, and then via a direct (and some-what more contemporary) proof.