Number theory, geometry and algebra - Dynamics-approx.jku.at

σ(n) = ∑ d|n d = S Id(n)—the sum of the divisors;
σ k (n) = ∑ d|n dk = S Id k(n) — the sum of the k-th powers of the divisors.
Lemma 2 If f is multiplicative, then so is S f .
Proof. We begin with the remark that each divisor d of a product mn of
two relatively prime numbers has a unique factorisation d 1 d 2 where d 1 |m,
d 2 |n.
For example, the divisors
of 12: 1,2,3,4,6,12, (43)
of 25: 1,5,25, (44)
of12×25 = 300 :1,2,3,4,6,12,5,10,15,20,30,60,25,50,75,100,150,300.
(45)
Hence
S f (mn) = ∑ d|mnf(d) =
∑
d 1 ,d 2 ,d 1 |m,d 2 |n
= ∑ d 1 |m
f(d 1 )f(d 2 ) (46)
f(d 1 ) ∑ d 2 |nf(d 2 ) (47)
= S f (m)S f (n). (48)
Example We calculate the values of τ and σ k . Since these functions are
multiplicative, it suffices to calculate their values for powers of primes. Since
the divisors of p α are {1,p,p 2 ,...,p α }, we have
Hence for n = p α 1
1 ...p αr
r
φ(p α ) = p α −p α−1 = p α (1− 1 p ) (49)
τ(p α ) = (α+1) (50)
α∑
σ k (p α ) = p ik = pk(α+1) −1
(k ≠ 0)
p−1
(51)
i=0
20
(52)