Question

Looking for a particular function...

Answer 1

The function I'm looking for I'll call \(f(n)\). It obeys this relationship to the zeta function:
\[\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}\]

Here's the relationship:
\[[\zeta(s)]^i = \sum_{n=1}^\infty \frac{f(n)}{n^s}\]

Answer 2

This is one sort of generalization of the Mobius function \(\mu(n)\) which obeys this relationship for anyone who's interested or curious why I'm interested in this. :P

\[[\zeta(s)]^{-1} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s}\]