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→Series relations: due to Ramanujan, add link to Rankin–Selberg method |
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: <math>\sum_{n=1}^\infty \frac{d(n)}{n^s} = \zeta^2(s)\quad\text{for}\quad s>1,</math>
and a [[Ramanujan]] identity{{sfnp|Hardy|Wright|2008|pp=334-337|loc=§17.8}}
:<math>\sum_{n=1}^\infty \frac{\sigma_a(n)\sigma_b(n)}{n^s} = \frac{\zeta(s) \zeta(s-a) \zeta(s-b) \zeta(s-a-b)}{\zeta(2s-a-b)}
which is a special case of the [[Rankin–Selberg method|Rankin–Selberg convolution]].
A [[Lambert series]] involving the divisor function is: {{sfnp|Hardy|Wright|2008|pp=338-341|loc=§17.10}}
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