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The sum of the odd divisors of n is $-\sum_{d|n}(-1)^{n/d}d$, and if n is even, then $$\sum_{d|n}(-1)^{n/d}d=2\sigma(n/2)-\sigma(n)$$

Could you give me some hints on that?

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2 Answers 2

up vote 3 down vote accepted

Hint : $d$ is an even divisors of $n$ if and only if $\frac{d}{2}$ is a divisors of $\frac{n}{2}$.


EDIT : Here's a complete proof. We shall prove two things : the sum of odd divisors of $n$ is given by the formula $-\sum_{d |n} (-1)^{n/d} d$, and if $n$ is even, then it's also equal to $\sigma(n)-2\sigma(n/2)$.

If $n$ is odd, this is obvious, so we're reduced to the case where $n$ is even. From the hint above, we get that the sum of even divisors of $n$ is $2\sigma(n/2)$. So the sum of odd divisors is $\sigma(n)-2\sigma(n/2)$. Finally, consider the sum

$$\sigma(n) + \sum_{d |n} (-1)^{d} \frac{n}{d} = \sum_{d |n} (1+(-1)^{d}) \frac{n}{d} = 2 \sum_{d |n, \ d \text{ even}} \frac{n}{d}$$

Changing the variable to $d' = d/2$, you get

$$\sigma(n) + \sum_{d |n} (-1)^{d} \frac{n}{d} = 2 \sum_{d' |n/2}\frac{n/2}{d'} = 2 \sigma(n/2)$$

Which concludes the proof. One a side note, you could perform the calculation using Dirichlet series, thus finding :

$$\eta(s) \zeta(s-1) = (1-2^{1-s}) \zeta(s) \zeta(s-1)$$

where $\eta(s) = \sum_{n \ge 1} \frac{(-1)^{n+1}}{n^s}$.

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It ought to be $\sum_{d|n}(-1)^{n/d}d$, rather than$\sum_{d |n} (-1)^{d} \frac{n}{d}$. –  Vladimir Nov 6 '11 at 13:43
    
@Z_Fang : both sums are equal (just change the variable from $d$ to $\frac{n}{d}$, which also runs through the divisors of $n$). Also note that I use this principle a second time to get $\sum_{d' | n/2} \frac{n/2}{d'} = \sigma(n/2)$. –  Joel Cohen Nov 6 '11 at 17:43
    
I was still wondering why it is obvious when n is odd. –  Vladimir Nov 6 '11 at 21:41
    
When $n$ is odd, every divisor is odd so the sign disappears and you end up with the plain sum of divisors (which is the sum of odd divisors). –  Joel Cohen Nov 6 '11 at 22:57

For the first part, you could try writing $n=2^km$, where $m$ is odd. Then the divisors of $n$ are the numbers $d,2d,4d,\dots,2^kd$ where $d$ runs through the odd divisors of $n$, which is to say, through the divisors of $m$. Now you can rewrite your sum to involve a sum on powers of 2.

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Err... How should I rewrite it? –  Vladimir Nov 6 '11 at 14:07
    
Well, your sum of $f(d)$ on all divisors $d$ of $n$ becomes a double sum of $f(2^rd)$ on all odd divisors $d$ of $n$ and on $r$ from 0 to $k$. –  Gerry Myerson Nov 6 '11 at 23:46

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