## Introduction to Analytic and Probabilistic Number Theory by Gerald Tenenbaum

oo), show that (a) pn, n log n (n oo) (b)E 4,1 log n x (x DO) 1

0, • Von Mangoldt's function A(n) := log p, if n = p" for some v > 1, if n is not a prime power. ) It is immediate from their definitions that S2 and w are additive, the former completely, the latter strongly. The case of the divisor function r(n) is less obvious. However, representing the divisors of n as all integers of the form d = H p% pin with 0 < ce p < VP (n,) for each prime p, we deduce that r(n) = ll(vp (n) ± 1). pin Thus we can state the following result. Theorem 1. The divisor function is multiplicative.

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