Difference between revisions of "Euler totient"
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where the notation $d | n$ indicates that $d$ is a [[divisor]] of $n$ and $\mu$ is the [[Möbius function]]. | where the notation $d | n$ indicates that $d$ is a [[divisor]] of $n$ and $\mu$ is the [[Möbius function]]. | ||
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where the notation $p | n$ indicates that $p$ is a prime that divides $n$. | where the notation $p | n$ indicates that $p$ is a prime that divides $n$. | ||
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where $\zeta$ is the [[Riemann zeta function]] and \mu is the [[Möbius function]], $e$ is the [[e | base of the exponential]] and the notation $d|n$ indicates that $d$ is any [[divisor]] of $n$. | where $\zeta$ is the [[Riemann zeta function]] and \mu is the [[Möbius function]], $e$ is the [[e | base of the exponential]] and the notation $d|n$ indicates that $d$ is any [[divisor]] of $n$. | ||
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| − | <strong>Proof:</strong> | + | <strong>Proof:</strong> █ |
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Revision as of 16:22, 4 October 2014
Euler's totient function (sometimes called Euler's $\phi$ function) is the function
Properties
Theorem: The function $\phi$ obeys the formula $$\phi(n) = \displaystyle\sum_{d|n} \mu(d) \dfrac{n}{d},$$ where the notation $d | n$ indicates that $d$ is a divisor of $n$ and $\mu$ is the Möbius function.
Proof: █
Theorem: The function $\phi$ obeys the formula $$\phi(n) = n \displaystyle\prod_{p | n} \left( 1 - \dfrac{1}{p} \right),$$ where the notation $p | n$ indicates that $p$ is a prime that divides $n$.
Proof: █
Theorem: The following formula holds: $$\phi(n) = n\lim_{s \rightarrow 1} \zeta(s) \displaystyle\sum_{d | n} \mu(d)(e^{\frac{1}{d}})^{s-1},$$ where $\zeta$ is the Riemann zeta function and \mu is the Möbius function, $e$ is the base of the exponential and the notation $d|n$ indicates that $d$ is any divisor of $n$.
Proof: █
