# Questions tagged [mobius-function]

Questions on the Möbius function μ(n), an arithmetic function used in number theory.

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### Is it true that this quadruple sum involving $f(x,y,z,p)=x^p+y^p−z^p$, is equal to the sequence $1,0,0,0,...$ only if $p=1$ or $p=2$?

Consider the sum: $$s(n) = \sum _{k=1}^n \left(\sum _{z=1}^n \left(\sum _{y=1}^n \left(\sum _{x=1}^n g(k)\left[\gcd (f(x,y,z,p),n)=k\right]\right)\right)\right)\label{1}\tag{1}$$ where the minimal ...
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### Möbius inversion and Möbius function as sum of cosines

Let $\mu (n)$ be the Möbius function. I want to prove the following formula: $$\mu (n)=\sum_{\substack{1\leq k \leq n\\ (k,n)=1}}\cos \frac{2k\pi}{n}.$$ Let $F(n)$ be the right hand side, then by ...
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### Perron's formula application - zeros of $L(s,\chi )$
After applying Perron's formula I have a complex integral involving something like $\sum _{n=1}^\infty \frac {\mu (n)}{n^s}e\left (\frac {an}{q}\right ).$ As usual ("usual" meaning e.g. ...