Difference between revisions of "Identity written as a sum of Möbius functions"

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(Created page with "==Theorem== The following formula holds for $|x|<1$: $$\displaystyle\sum_{k=1}^{\infty} \dfrac{\mu(k)x^k}{1-x^k} = x,$$ where $\mu$ denotes the Möbius function. ==Proof...")
 
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==References==
 
==References==
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Reciprocal of Riemann zeta as a sum of Möbius function for Re(z) greater than 1|next=}}: 24.3.1 B
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Reciprocal of Riemann zeta as a sum of Möbius function for Re(z) greater than 1|next=}}: 24.3.1 I.B.
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
 
[[Category:Unproven]]
 
[[Category:Unproven]]

Revision as of 01:23, 22 June 2016

Theorem

The following formula holds for $|x|<1$: $$\displaystyle\sum_{k=1}^{\infty} \dfrac{\mu(k)x^k}{1-x^k} = x,$$ where $\mu$ denotes the Möbius function.

Proof

References