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.

#NumberTheory

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Comments

Both sums diverges.

Aareyan Manzoor - 5 years, 3 months ago

THe first sum diverges Link

The second sum follows suit.

Julian Poon - 5 years, 3 months ago

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 - 5 years, 3 months ago

Diverges means doesnt converge to a specific finite value, so it has to be eother one.

Aareyan Manzoor - 5 years, 3 months ago

@Aareyan Manzoor – Ok, so it diverges

Julian Poon - 5 years, 3 months ago

@Julian Poon – @Julian Poon , @Aareyan Manzoor any modifications ?

A Former Brilliant Member - 5 years, 3 months ago

Till some point I also thought like this , but later , I left it as I thought it may be wrong.

A Former Brilliant Member - 5 years, 3 months ago

@A Former Brilliant Member – THere is a high possibility my steps aren't justified, since I have assumed the sum converged.

Julian Poon - 5 years, 3 months ago

@Julian Poon – @Julian Poon , @Aareyan Manzoor I'm waiting for your reply

A Former Brilliant Member - 5 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$