Help me prove
$$ \sum_{n=1}^\infty \frac{\sigma_a(n)\sigma_b(n)}{n^s} = \frac{\zeta(s) \zeta(sa) \zeta(sb) \zeta(sab)}{\zeta(2sab)} $$
Help me prove $$ \sum_{n=1}^\infty \frac{\sigma_a(n)\sigma_b(n)}{n^s} = \frac{\zeta(s) \zeta(sa) \zeta(sb) \zeta(sab)}{\zeta(2sab)} $$ 

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The functions involved are "multiplicative". Therefore it is enough to prove the "local result":the same sum running over the nonnegative powers of the same prime on the left anf the product of local factors of Riemann's zeta function on the right. To deal with the resulting power series on the left, use the formula for the geometric series. 


Suggest you look at page 59 of “Ramanujan” by G.H.Hardy, 3rd edition published by Chelsea Publishing Company 1978. 

