Difference between revisions of "Identity written as a sum of Möbius functions"
From specialfunctionswiki
(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...") |
(No difference)
|
Revision as of 01:22, 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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous): 24.3.1 B
