The Sum of the Odd Divisors of n 
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?
 A: 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 $\eta(s) = \sum_{n \ge 1} \frac{(-1)^{n+1}}{n^s}$.
A: 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. 
