Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due
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.]