Number Theory Level 5

\[\large \sum_{n \text{ cube-free}} \dfrac{(-2)^{2\omega(n)-\Omega(n)}}{n^2}\]

If the sum above can be expressed in the form of \( \dfrac a{b \pi^m} \), where \(a,b\) and \(m\) are positive integers with \(a,b\) coprime, find \(a+b+m\).


  • \(n\) runs through all positive integers.

  • \(\omega(n)\) denotes the number of distinct prime divisors of \(n\).

  • \(\Omega(n)\) counts the prime divisors of \(n\) with multiplicity.


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