Difference between revisions of "Euler totient"
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=Properties= | =Properties= | ||
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| + | <strong>Theorem:</strong> 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]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> proof goes here █ | ||
| + | </div> | ||
| + | </div> | ||
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
<strong>Theorem:</strong> The function $\phi$ obeys the formula | <strong>Theorem:</strong> The function $\phi$ obeys the formula | ||
Revision as of 16:21, 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: proof goes here █
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: proof goes here █
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: proof goes here █
