Difference between revisions of "Identity written as a sum of Möbius functions"
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==Theorem== | ==Theorem== | ||
The following formula holds for $|x|<1$: | The following formula holds for $|x|<1$: | ||
| − | $$\displaystyle\sum_{k=1}^{\infty} \dfrac{\mu(k)x^k}{1-x^k} | + | $$x=\displaystyle\sum_{k=1}^{\infty} \dfrac{\mu(k)x^k}{1-x^k},$$ |
where $\mu$ denotes the [[Möbius function]]. | where $\mu$ denotes the [[Möbius function]]. | ||
Revision as of 01:25, 22 June 2016
Theorem
The following formula holds for $|x|<1$: $$x=\displaystyle\sum_{k=1}^{\infty} \dfrac{\mu(k)x^k}{1-x^k},$$ where $\mu$ denotes the Möbius function.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous): 24.3.1 I.B.
