# Euler totient

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 $n$.

# 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)

Euler totient function made easy (17 December 2017)

# References

- 1964: Milton Abramowitz and Irene A. Stegun:
*Handbook of mathematical functions*... (previous) ... (next): $24.3.2 \mathrm{I}.A.$

**Number theory functions**