Number Theory (MA3Z7)

Problem Sheet VIII

1. Let d — the sum of the divisors of n.

(i) Show that

(ii) Apply Theorem 5.4 to prove that

(iii) Deduce that σ(n) has average order

2. A natural number is cubefree if it contains no cubed prime factor. By mimicking the squarefree case:

(i) show that for s > 1,

(ii) show that the number of cubefree integers up to N is, asymp-totically,

3. (i) Use the relation where k(n) = µ(m) if n = m2 and zero otherwise, to show that

(ii) Let w(n) denote the number of distinct prime factors of n.

Deduce from (i) that the average order of 2 w(n) is

[You may want to look at Problem Sheet VI, Q4.]