R.T.P. σ(p^n)=(p^(n+1)-1)/(p-1). (where σ denotes the divisor function) For a prime p.

Question

I need to prove that for a prime p, σ(p^n) is (p^(n+1)-1)/(p-1).

E.g.

σ(3^3) is (3^(3+1)-1)/(3-1).

=3^(4)-1/2

=80/2

=40

Therefore σ(3^3)=40.

Let me know for any suggestions,proofs or references.

Answer 1

The divisors of $p^n$ are $1, p, p^2, \cdots, p^n$.

$$\sigma(p^n)=1+p+p^2+\cdots+p^n=\frac{p^{n+1}-1}{p-1}$$