1. ## Sigma Function Help

Sigma Function

σ: ℕ ->ℕ

n
Σ k = σ (n)
k|n

1. Is the Sigma Fctn an injection?
2. Is it a surjection?
3. Can you find a formula for the sigma fctn which allows to calculate the sum of divisors of n based on the decomposition of n into a product of prime numbers?

Thanks guys!

2. Originally Posted by suissa
n
Σ k = σ (n)
k|n

1. Is the Sigma Fctn an injection?
2. Is it a surjection?
3. Can you find a formula for the sigma fctn which allows to calculate the sum of divisors of n based on the decomposition of n into a product of prime numbers?
1) Consider $\displaystyle n=15,23$.
2) Can it be that $\displaystyle \sigma (n) = 2$?
3) If $\displaystyle n=p_1^{a_1}...p_n^{a_n}$ then $\displaystyle \sigma (n) = \frac{p_1^{a_1}-1}{p_1-1}\cdot ... \cdot \frac{p_n^{a_n}-1}{p_n-1}$.

1. Is the Sigma Fctn an injection?
No. Look for a counter-example, it shouldn't take too long.

2. Is it a surjection?
No. look for another counter-example

3. Can you find a formula for the sigma fctn which allows to calculate the sum of divisors of n based on the decomposition of n into a product of prime numbers?
Yes.
If n = $\displaystyle \prod_{i=0}^ma_i^{k_i}$ where all a are prime,
then $\displaystyle \sigma(n) = \prod_{i=0}^m\sum_{j_i=0}^{k_i}a_i^{j_i}$
This can be simplified using the formula for geometric series

4. I'm sorry I don't understand 1 and 2.
Thanks a lot for the reply though!

5. Oh I understand it.