Difference between revisions of "Euler totient"
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| − | Euler's totient function $\phi$ (not to be confused with the [[Euler phi]]) is defined | + | Euler's totient function $\phi \colon \{1,2,3,\ldots\} \rightarrow \{1,2,3,\ldots\}$ (not to be confused with the [[Euler phi]]) is defined so that $\phi(n)$ equals the number of positive integers less than or equal to $n$ that are [[relatively prime]] to $n$. |
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Revision as of 23:51, 10 December 2016
Euler's totient function $\phi \colon \{1,2,3,\ldots\} \rightarrow \{1,2,3,\ldots\}$ (not to be confused with the Euler phi) is defined so that $\phi(n)$ equals the number of positive integers less than or equal to $n$ that are relatively prime to $n$.
Graph of $\phi$ to $3500$.
Properties
Sum of totient equals zeta(z-1)/zeta(z) for Re(z) greater than 2
Sum of totient equals z/((1-z) squared)
Product representation of totient
Euler totient is multiplicative
Videos
Euler's Phi Function (6 October 2010)
Euler's Totient Function (14 October 2011)
Euler Totient Theorem, Fermat Little Theorems (19 December 2011)
Möbius and Euler totient functions (2 September 2012)
Prime Factorisation and Euler Totient Function Part 14 (16 April 2014)
Application of Euler Totient Function Part 16 (17 April 2014)
Euler's Totient Function: what it is and how it works (4 July 2014)
Euler's Totient Theorem: What is Euler's Totient Theorem and Why is it useful? (6 July 2014)
Euler's Totient Function | How To Find Totient Of A Number Using Euler's Product Formula (7 July 2014)
03 Modern cryptography 08 Euler's totient function (23 September 2014)
Euler's totient function (8 January 2015)
Euler Totient (phi) Function Examples (Part 1) (27 November 2016)
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $24.3.2 \mathrm{I}.A.$
