I would like to prove that

$\displaystyle \sum_{n = 1}^\infty \frac{\lambda_a}{n^s} = \frac{\zeta(s) \zeta(2s-2a)}{\zeta(s-a)} $

where

$\displaystyle \lambda_a(n) = \sum_{d \vert n} d^a \cdot \lambda(d) $

and

$\displaystyle \lambda(n) $ is Liouville's function, defined as

$\displaystyle \lambda(1) = 1 \mbox{, and if } n=p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} \mbox{ then } \lambda(n) = (-1)^{a_1 + a_2 + \cdots + a_k} $

Using multiplication of Dirichlet series, I think I can show that

$\displaystyle \sum_{n = 1}^\infty \frac{\lambda (n)}{n^s} = \frac{\zeta(2s)}{\zeta(s)} $

and that

$\displaystyle \sum_{n = 1}^\infty \frac{\sum_{d \vert n} d^a}{n^s} = \zeta(s-a)\zeta(s) $

but I cant see how (or if) this helps.

Any help or advice would be appreciated.