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Divergent Möbius series

Let \mu denote the Möbius function. Prove that

    \[\lim_{N \rightarrow +\infty} \sum_{n=1}^{N} \frac{\left| \mu (n) \right|}{n} = +\infty\]

Solution

Summing only over primes , where |\mu(p)|=1 ,  we have that

    \[\lim_{N \rightarrow +\infty} \sum_{n=1}^{N} \frac{\left| \mu (n) \right|}{n} \geq \sum_{p \in \mathcal{P}} \frac{1}{p} = +\infty\]

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