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Möbius Mania

\[\sum_{n=1}^{\infty} \mu (n)\]

1] Prove/Disprove , that the sum mentioned above converges.

Also ,

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

2] Prove / Disprove , the sum mentioned above converges.

Note by Chinmay Sangawadekar
9 months, 1 week ago

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THe first sum diverges Link

The second sum follows suit. Julian Poon · 9 months, 1 week ago

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@Julian Poon EDIT: My new thinking is that both sums does not converge or diverge since it's value fluctuates.

Either ways, here's my approach:

Let \(f\) be a completely multiplicative function.

\[\sum_{n>0}f(n)=\left[\prod_{p \text{ is prime}}(1-f(p))\right]^{-1}\]

Through euler product.

Expanding the product gives

\[\prod_{p \text{ is prime}}(1-f(p))=\sum_{n>0}\mu(n)f(n)\]

Putting it all together

\[\sum_{n>0}f(n)=\left[\sum_{n>0}\mu(n)f(n)\right]^{-1}\]

Substituting \(f=1\) gives

\[\sum_{n>0}1=\left[\sum_{n>0}\mu(n)\right]^{-1}\]

\[\sum_{n>0}\mu(n)=0\]

Of course there is a lot of hand waving here. Julian Poon · 9 months, 1 week ago

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@Julian Poon Diverges means doesnt converge to a specific finite value, so it has to be eother one. Aareyan Manzoor · 9 months, 1 week ago

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@Aareyan Manzoor Ok, so it diverges Julian Poon · 9 months, 1 week ago

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@Julian Poon @Julian Poon , @Aareyan Manzoor any modifications ? Chinmay Sangawadekar · 9 months, 1 week ago

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@Julian Poon Till some point I also thought like this , but later , I left it as I thought it may be wrong. Chinmay Sangawadekar · 9 months, 1 week ago

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@Chinmay Sangawadekar THere is a high possibility my steps aren't justified, since I have assumed the sum converged. Julian Poon · 9 months, 1 week ago

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@Julian Poon @Julian Poon , @Aareyan Manzoor I'm waiting for your reply Chinmay Sangawadekar · 9 months, 1 week ago

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Both sums diverges. Aareyan Manzoor · 9 months, 1 week ago

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