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In Number Theory , the totient \phi(''n'') of a Positive Integer ''n'' is defined to be the number of positive integers less than or equal to ''n'' and Coprime to ''n''.
For example, \phi(8) = 4 since the four numbers 1, 3, 5 and 7 are coprime to 8.
The Function \phi so defined is the totient function.
The totient is usually called the Euler totient or '''Euler's totient''', after the Swiss mathematician Leonhard Euler , who studied it.
The totient function is also called Euler's phi function or simply the '''phi function''', since the letter Phi (\phi) is so commonly used for it. The '''cototient''' of ''n'' is defined as ''n'' − \phi(''n'').

The totient function is important mainly because it gives the size of the multiplicative Group of integers Modulo ''n''. More precisely, \phi(''n'') is the order of the group of Unit s of the Ring \mathbb{Z}/n\mathbb{Z}. This fact, together with Lagrange's Theorem , provides a proof for Euler's Theorem .


COMPUTING EULER'S FUNCTION


It follows from the definition that \phi(1) = 1, and \phi(''n'') is (''p'' − 1)''p''''k''−1 when ''n'' is the ''k''th power of a between ''A''x''B'' and ''C'', via the Chinese Remainder Theorem .) The value of \phi(''n'') can thus be computed using the Fundamental Theorem Of Arithmetic : if

n


where the ''p''''j'' are distinct Primes ,
then

: arphi(n)=(p_{1}-1)p_{1}^{k_{1}-1} \cdots (p_{r}-1)p_{r}^{k_{r}-1}

This last formula is a Euler Product and is often written as
  { Class "wikitable"
  The Number <math>\phi</math>(''n'') Is Also Equal To The Number Of Possible Generators Of The "http://wwwinformationdelightinfo/encyclopedia/entry/Vrhbosna/cyclic_group" class="copylinks">Cyclic Group ''C''<sub>''n''</sub> (and therefore also to the degree of the Cyclotomic Polynomial <math>\phi</math><sub>''n''</sub>) Since every element of ''C''<sub>''n''</sub> generates a cyclic Subgroup and the subgroups of ''C''<sub>''n''</sub> are of the form ''C''<sub>''d''</sub> where ''d'' Divides ''n'' (written as ''d''''n''), we get


for n > 2, where γ is Euler's Constant ,

:
arphi(n) \ge \sqrt{ rac {n} {2} }
for ''n'' > 0,

and
:
arphi(n) \ge \sqrt{n}
for ''n'' > 6.

For prime ''n'', clearly \phi(''n'') = ''n''-1. For Composite ''n'' we have
:
arphi(n) \le n-\sqrt{n}
(for composite ''n'').

For all n>1 :

0< rac{ arphi (n)}{n}<1

For randomly large n, these bounds still cannot be improved, or to be more precise :

lim inf rac{ arphi (n)}{n}=0 and lim sup rac{ arphi (n)}{n}=1

A pair of inequalities combining the \phi function and the σ Divisor Function are:

:
rac {6 n^2} {\pi^2} < arphi(n) \sigma(n) < n^2
for ''n'' > 1.


SEE ALSO




REFERENCES