# Sum of Positive Divisors Function

• Jul 23rd 2013, 03:34 PM
younghorsey
Sum of Positive Divisors Function

Let $\displaystyle n \in Z$ with $\displaystyle n > 0$

Prove that:

$\displaystyle (\sum_{d|n, d>0}\upsilon(d))^2 =$
$\displaystyle \sum_{d|n, d>0}(\upsilon(d))^3$

[Hint: It suffices to prove the equation above for powers of prime numbers (Why?)]
• Jul 23rd 2013, 06:41 PM
chiro
Re: Sum of Positive Divisors Function
Hey younghorsey.

What is v(d)? Is this some kind of special analytic number theory function?
• Jul 24th 2013, 07:26 AM
younghorsey
Re: Sum of Positive Divisors Function
Hey chiro.

$\displaystyle \upsilon(n) = \sum_{d|n, d>0} 1$
• Jul 24th 2013, 05:40 PM
chiro
Re: Sum of Positive Divisors Function
I don't know enough number theory to help you on this one.

What I would recommend you do is collect as many identities as possible and put them all in the one place: this will help you see the connections a lot quicker and should help you solve your problem quicker than say a less structured approach.

If you have all the identities and theorems on a few pages the connections should pop out.
• Jul 26th 2013, 02:25 PM
johng
Re: Sum of Positive Divisors Function
Hi,
The attachment provides a proof. I've left the proofs of several relevant facts to you.

Attachment 28905