1. PANORAMA OF ARITHMETIC FUNCTIONS 7

From (1.21) we derive by induction in k that Λk(n) 0 and Λk(n) is supported

on positive integers having at most k distinct prime divisors. Moreover we get by

(1.20) that

(1.22) 0 Λk(n) (log

n)k.

Exercise. Prove the formula

(1.23) Λk(mn) =

0 j k

k

j

Λj (m)Λk−j(n).

Exercise. Prove the formula

(1.24)

p

1

ps

=

∞

1

μ(n)

n

log ζ(ns), if s 1.