Given n ≥ 1, let σs(n) denote the sum of the sth powers of the positive divisors of n; tha...

Given n ≥ 1, let σs(n) denote the sum of the sth powers of the positive divisors of n; that is,

Verify the following:

(a) σ0 = τ and σ1 = σ.

(b) σs is a multiplicative function.

[Hint: The function f, defined by f(n)= ns, is multiplicative.]

(c) If  is the prime factorization of n, then