Fundamental Thm of Arithemetic

[1+1=1]

Wow, it has been awhile! China has been fun, but now it is time to get back to the states and also math work! Here is a problem that has been giving me some problems. It reads: \prod from i=1 to n, pi^ai for each i is the canonical representation of a, deduce a formula for the sum of squares of the positive divisors of a. I know what the canonical representation is, so would i just plug in numbers for n, and then from the output just make up a \sum formula? Could anyone provide guidance?

[shmoe]

Hi, are you familiar with the formula for the sum of the divisors of a? (not the squares). If so, this is a very closely related problem.

If not, here are the basic steps. Let f(a)=sum of squares of divisors.

1)find a formula when a is a prime power, that is if a=p^b, where p is prime, what is f(p^b)? If you need another hint, what are the divisors of p^b?

2)if m and n are relatively prime, show f(nm)=f(n)f(m), that is, f is multiplicative. If you have trouble with the general case here, try a simplified form first, where m and n are prime powers. This should help you see how to get the divisors of nm from the divisors of m and the divisors of n.

3)Combine the above to get a feneral formula for f(a) in terms of it's prime factorization.