View source for Pionium

{{for|other functions named after Euler|List of topics named after Leonhard Euler}}
[[Image:EulerPhi.svg|thumb|The first thousand values of <math>\varphi(n)</math>]]
In [[number theory]], '''Euler's totient''' or '''phi function''', φ(''n''), is an [[arithmetic function]] that counts the [[totative]]s of ''n'', that is, the positive integers less than or equal to ''n'' that are [[relatively prime]] to ''n''.  Thus if ''n'' is a [[positive integer]], then φ(''n'') is the number of integers ''k'' in the range 1 ≤  ''k'' ≤  ''n'' for which [[Greatest common divisor|gcd]](''n'', ''k'') = 1.<ref>{{harvtxt|Long|1972|p=85}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=72}}</ref> The totient function is a [[multiplicative function]], meaning that if two numbers ''m'' and ''n'' are relatively prime (to each other), then φ(''mn'')&nbsp;=&nbsp;φ(''m'')φ(''n'').<ref>{{harvtxt|Long|1972|p=162}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=80}}</ref>

For example let ''n'' = 9. Then gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9. The other six numbers in the range 1 ≤  ''k'' ≤  9, that is, 1, 2, 4, 5, 7 and 8, are relatively prime to 9. Therefore, φ(9) = 6. As another example, φ(1) = 1 since gcd(1, 1) = 1.

The totient function is important mainly because it gives the [[order (group theory)|order]] of the [[multiplicative group of integers modulo n|multiplicative group of integers modulo ''n'']] (the [[Multiplicative group of integers modulo n|group]] of [[unit (ring theory)|unit]]s of the [[ring (algebra)|ring]] <math>\mathbb{Z}/n\mathbb{Z}</math>). See [[#Euler's theorem|Euler's theorem]].<br>The totient function also plays a key role in the definition of the [[#The RSA cryptosystem|RSA]] encryption system.

==History, terminology, and notation==

[[Leonhard Euler]] introduced the function in 1760.<ref>L. Euler (1760) "Theoremata arithmetica nova methodo demonstrata" (An arithmetic theorem proved by a new method), ''Novi commentarii academiae scientiarum imperialis Petropolitanae'' (New Memoirs of the St. Petersburg Imperial Academy of Sciences), '''8''' : 74-104.  Available on-line in:  [[Ferdinand Rudio]], ed., ''Leonhardi Euleri Commentationes Arithmeticae'', volume 1, in: ''Leonhardi Euleri Opera Omnia'', series 1, volume 2 (Leipzig, Germany:  B.G. Teubner, 1915), [http://gallica.bnf.fr/ark:/12148/bpt6k6952c/f571.image pages 531-555].  On page 531, Euler defines ''n'' as the number of integers that are smaller than ''N'' and relatively prime to ''N'' (… aequalis sit multitudini numerorum ipso N minorum, qui simul ad eum sint primi, …), which is the totient function, φ(N).</ref><ref>Sandifer, p. 203</ref><ref>Graham et al. p. 133 note 111</ref> The standard notation<ref>Sandifer, p. 203</ref><ref>Both  <math>\varphi(n)\;</math> and <math>\phi(n)\;</math> are seen in the literature. These are two forms of the lower-case Greek letter [[phi]]</ref> φ(''n'') is from [[Gauss]]' 1801 treatise ''[[Disquisitiones Arithmeticae]]''.<ref>Gauss, DA art. 38</ref> Thus, it is usually called '''Euler's phi function''' or simply the '''phi function'''.

In 1879 [[James Joseph Sylvester|J. J. Sylvester]] coined the term '''totient''' for this function,<ref>J. J. Sylvester (1879) "On certain ternary cubic-form equations," ''American Journal of Mathematics'', '''2''' : 357-393; Sylvester coins the term "totient" on [http://books.google.com/books?id=-AcPAAAAIAAJ&pg=PA361#v=onepage&q&f=false page 361].</ref> so it is also referred to as the '''totient function''', the '''Euler totient''', or '''Euler's totient'''. 
[[Jordan's totient function|Jordan's totient]] is a generalization of Euler's.

The '''cototient''' of ''n'' is defined as ''n'' – φ(''n''), i.e., the number of positive integers less than or equal to ''n'' that are divisible by at least one [[prime number|prime]] that also divides ''n''.

==Computing Euler's function==

There are several formulae for the totient.

===Euler's product formula===

It states
:<math>
\varphi(n) =n \prod_{p\mid n} \left(1-\frac{1}{p}\right),
</math>
where the product is over the distinct [[prime number]]s dividing n. (The notation is described in the article [[Arithmetical function#Notation|Arithmetical function]].)

The proof of Euler's product formula depends on two important facts.

====φ(''n'') is multiplicative====

This means that if gcd(''m'', ''n'') = 1, then φ(''mn'') = φ(''m'') φ(''n''). <br>(Sketch of proof: let ''A'', ''B'', ''C'' be the sets of residue classes modulo-and-coprime-to ''m'', ''n'', ''mn'' respectively; then there is a [[bijection]] between ''A''&nbsp;&times;&nbsp;''B'' and ''C'', by the [[Chinese remainder theorem]].)

====φ(''p''<sup>''k''</sup>) = ''p''<sup>''k''</sup> − ''p''<sup>''k'' − 1</sup> = ''p''<sup>''k'' − 1</sup>(''p'' − 1)====

That is, if ''p'' is prime and ''k'' ≥ 1 then 

:<math>\varphi(p^k) = p^k -p^{k-1} =p^{k-1}(p-1) = p^k \left( 1 - \frac{1}{p} \right).</math>

''Proof:'' Since ''p'' is a prime number the only possible values of gcd(''p''<sup>''k''</sup>, ''m'') are 1, ''p'', ''p''<sup>2</sup>, ..., ''p''<sup>''k''</sup>, and the only way for gcd(''p''<sup>''k''</sup>, ''m'') to not equal 1 is for ''m'' to be a multiple of ''p''. The multiples of ''p'' that are less than or equal to ''p''<sup>''k''</sup> are ''p'', 2''p'', 3''p'', ..., ''p''<sup>''k'' − 1</sup>''p'' = ''p''<sup>''k''</sup>, and there are ''p''<sup>''k'' − 1</sup> of them. Therefore the other ''p''<sup>''k''</sup> − ''p''<sup>''k'' − 1</sup> numbers are all relatively prime to ''p''<sup>''k''</sup>.

''Proof of the formula:''
The [[fundamental theorem of arithmetic]] states that if ''n'' > 1 there is a unique expression for ''n'',

:<math>n = p_1^{k_1} \cdots p_r^{k_r}, </math>

where ''p''<sub>1</sub> < ''p''<sub>2</sub> < ... < ''p''<sub>''r''</sub> are [[prime number]]s and each ''k''<sub>''i''</sub> ≥  1. (The case ''n'' = 1 corresponds to the empty product.)

Repeatedly using the multiplicative property of φ and the formula for φ(''p''<sup>''k''</sup>) gives

:<math>
\begin{align} 
\varphi(n)
&= \varphi(p_1^{k_1}) \varphi(p_2^{k_2}) \cdots\varphi(p_r^{k_r})\\

&=  p_1^{k_1} \left(1- \frac{1}{p_1} \right) p_2^{k_2} \left(1- \frac{1}{p_2} \right) \cdots p_r^{k_r} \left(1- \frac{1}{p_r} \right)\\

&= p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r} \left(1- \frac{1}{p_1} \right) \left(1- \frac{1}{p_2} \right) \cdots \left(1- \frac{1}{p_r} \right)\\

&=n \left(1- \frac{1}{p_1} \right)\left(1- \frac{1}{p_2} \right) \cdots\left(1- \frac{1}{p_r} \right).
\end{align}
</math>

This is Euler's product formula.

====Example====

:<math>\varphi(36)=\varphi\left(2^2 3^2\right)=36\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)=36\cdot\frac{1}{2}\cdot\frac{2}{3}=12.</math>

In words, this says that the distinct prime factors of 36 are 2 and 3; half of the thirty-six integers from 1 to 36 are divisible by 2, leaving eighteen; a third of those are divisible by 3, leaving twelve numbers that are coprime to 36.  And indeed there are twelve positive integers that are coprime with 36 and lower than 36: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, and 35.

===Fourier transform===

The totient is the [[discrete Fourier transform]] of the [[Greatest common divisor|gcd]], evaluated at 1: &nbsp; ({{harvtxt|Schramm|2008}}) 
:<math>\mathcal{F} \left \{ \mathbf{x} \right \}[m] = \sum\limits_{k=1}^n x_k \cdot e^{{-2\pi i}\tfrac{mk}{n}}, \mathbf{x} = \left \{ \gcd(k,n) \right \} \quad\text{for}\, k \in \left \{1 \dots n \right \}
</math>
:<math>\varphi (n)=\mathcal{F} \left \{ \mathbf{x} \right \}[1] =\sum\limits_{k=1}^n \gcd(k,n) e^{{-2\pi i}\tfrac{k}{n}}. </math>
The real part of this formula is
:<math>
\varphi (n)=\sum\limits_{k=1}^n \gcd(k,n) \cos {2\pi\frac{k}{n}}.
</math>
Note that unlike the other two formulae (the Euler product and the divisor sum) this one does not require knowing the factors of ''n''.  However, it does involve the calculation of the greatest common divisor of ''n'' and every positive integer less than ''n'', which would suffice to provide the factorization anyway.

===Divisor sum===

Euler's classical formula<ref>Hardy & Wright, thm. 63, note to § 5.5</ref><ref>Gauss, DA, art 39</ref> 
:<math>
\sum_{d\mid n}\varphi(d)=n,
</math>
where the sum is over all positive divisors ''d'' of ''n'', can be proven in several ways. (see [[Arithmetical function#Notation|Arithmetical function]] for notational conventions.)

One way is to note that φ(''d'') is also equal to the number of possible generators of the [[cyclic group]] ''C''<sub>''d''</sub>; specifically, if ''C''<sub>''d''</sub> = <''g''>, then ''g''<sup>''k''</sup> is a generator for every ''k'' coprime to ''d''. Since every element of ''C''<sub>''n''</sub> generates a cyclic [[subgroup]], and all φ(''d'') subgroups of ''C''<sub>''d''</sub> ≤ ''C''<sub>''n''</sub> are generated by some element of ''C''<sub>''n''</sub>, the formula follows.<ref>Gauss, DA art. 39, arts. 52-54</ref> In the article [[Root of unity#Elementary facts|Root of unity]] Euler's formula is derived by using this argument in the special case of the multiplicative group of the ''n''th roots of unity.

This formula can also be derived in a more concrete manner.<ref>Graham et al. pp. 134-135</ref> Let ''n'' = 20 and consider the fractions between 0 and 1 with denominator 20:
:<math>
\tfrac{ 1}{20},\,\tfrac{ 2}{20},\,\tfrac{ 3}{20},\,\tfrac{ 4}{20},\,
\tfrac{ 5}{20},\,\tfrac{ 6}{20},\,\tfrac{ 7}{20},\,\tfrac{ 8}{20},\,
\tfrac{ 9}{20},\,\tfrac{10}{20},\,\tfrac{11}{20},\,\tfrac{12}{20},\,
\tfrac{13}{20},\,\tfrac{14}{20},\,\tfrac{15}{20},\,\tfrac{16}{20},\,
\tfrac{17}{20},\,\tfrac{18}{20},\,\tfrac{19}{20},\,\tfrac{20}{20}
</math>
Put them into lowest terms:
:<math>
\tfrac{ 1}{20},\,\tfrac{ 1}{10},\,\tfrac{ 3}{20},\,\tfrac{ 1}{ 5},\,
\tfrac{ 1}{ 4},\,\tfrac{ 3}{10},\,\tfrac{ 7}{20},\,\tfrac{ 2}{ 5},\,
\tfrac{ 9}{20},\,\tfrac{ 1}{ 2},\,\tfrac{11}{20},\,\tfrac{ 3}{ 5},\,
\tfrac{13}{20},\,\tfrac{ 7}{10},\,\tfrac{ 3}{ 4},\,\tfrac{ 4}{ 5},\,
\tfrac{17}{20},\,\tfrac{ 9}{10},\,\tfrac{19}{20},\,\tfrac{ 1}{ 1}
</math>
First note that all the divisors of 20 are denominators. And second, note that there are 20 fractions.<br> Which fractions have 20 as denominator? The ones whose numerators are relatively prime to 20 <math> (\tfrac{ 1}{20},\,\tfrac{ 3}{20},\,\tfrac{ 7}{20},\,\tfrac{ 9}{20},\,\tfrac{ 11}{20},\,\tfrac{13}{20},\,\tfrac{17}{20},\,\tfrac{19}{20}).</math> <br>By definition this is φ(20) fractions. 
<br>Similarly, there are φ(10) = 4 fractions with denominator 10  <math> (\tfrac{ 1}{10},\,\tfrac{ 3}{10},\,\tfrac{ 7}{10},\,\tfrac{ 9}{10}),</math> φ(5) = 4 fractions with denominator 5  <math> (\tfrac{ 1}{5},\,\tfrac{ 2}{5},\,\tfrac{ 3}{5},\,\tfrac{ 4}{5}),</math> and so on. Since the same argument works for any number, not just 20, the formula is established.

[[Möbius inversion]] gives

:<math>\varphi(n)
=\sum_{d\mid n} d \cdot \mu\left(\frac{n}{d} \right) 
=n\sum_{d\mid n} \frac{\mu (d)}{d}  
,
</math>

where μ is the [[Möbius function]].

This formula may also be derived from the product formula by multiplying out &nbsp; <math> \prod_{p\mid n} \left(1-\frac{1}{p}\right) </math> &nbsp; to get &nbsp; <math> \sum_{d\mid n} \frac{\mu (d)}{d}. </math>

===Riemann zeta function limit===

For <math> n>1 </math> the Euler totient function can be calculated as a limit involving the Riemann zeta function:{{citation needed|reason=this formula is not obvious at all, a reference is needed|date=February 2014}}

:<math> \varphi(n)=n\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \mu(d)(e^{1/d})^{(s-1)}</math>

where 

<math> \zeta(s)</math>
is the [[Riemann zeta function]],
<math> \mu</math> is the [[Möbius function]],
<math> e</math> is [[e (mathematical constant)]],
and <math> d</math> is a divisor.

==Some values of the function==

The first 99 values {{OEIS|A000010}} are shown in the table and graph below:
[[File:EulerPhi100.PNG|thumb|Graph of the first 100 values]]
{| class="wikitable"
|-
! <math>\varphi(n)</math>
! +0 || +1 || +2 || +3 || +4 || +5 || +6 || +7 || +8 || +9
|- 
! 0+ 
| &nbsp; || 1 || 1 || 2 || 2 || 4 || 2 || 6 || 4 || 6
|-
!10+ 
|  4 || 10 || 4 || 12 ||  6 ||  8 ||  8 || 16 ||  6 || 18 
|- 
!20+ 
|  8 || 12 || 10 || 22 || 8 || 20 || 12 || 18 || 12 || 28 
|-
!30+ 
|  8 || 30 || 16 || 20 || 16 || 24 || 12 || 36 || 18 || 24
|-
!40+
| 16 || 40 || 12 || 42 || 20 || 24 || 22 || 46 || 16 || 42
|-
!50+
| 20 || 32 || 24 || 52 || 18 || 40 || 24 || 36 || 28 || 58
|-
!60+ 
| 16 || 60 || 30 || 36 || 32 || 48 || 20 || 66 || 32 || 44
|-
!70+ 
| 24 || 70 || 24 || 72 || 36 || 40 || 36 || 60 || 24 || 78
|-
!80+
| 32 || 54 || 40 || 82 || 24 || 64 || 42 || 56 || 40 || 88
|-
!90+ 
| 24 || 72 || 44 || 60 || 46 || 72 || 32 || 96 || 42 || 60
|}

The top line in the graph, ''y'' = ''n'' − 1, is a true [[upper bound]]. It is attained whenever ''n'' is prime.<br>
The lower line, ''y'' ≈ 0.267''n'' which connects the points for  ''n'' = 30, 60, and 90 is misleading. If the plot were continued, there would be points below it. <br>(Examples: for ''n''&nbsp;=&nbsp;210&nbsp;=&nbsp;7×30, φ(''n'')&nbsp;≈&nbsp;0.229&nbsp;''n''; for ''n''&nbsp;=&nbsp;2310&nbsp;=&nbsp;11×210  φ(''n'')&nbsp;≈&nbsp;0.208&nbsp;''n''; and for ''n''&nbsp;=&nbsp;30030&nbsp;=&nbsp;13×2310 φ(''n'')&nbsp;≈&nbsp;0.192 ''n''.) <br>In fact, there is no [[lower bound]] that is a straight line of positive slope; no matter how gentle the slope of a line is, there will eventually be points of the plot below the line.
{{clear}}

==Euler's theorem==

{{main|Euler's theorem}}

This states that if ''a'' and ''n'' are relatively prime then

:<math> a^{\varphi(n)} \equiv 1\mod n.\,</math>

The special case where ''n'' is prime is known as [[Fermat's little theorem]]

This follows from [[Lagrange's theorem (group theory)|Lagrange's theorem]] and the fact that φ(''n'') is the [[order (group theory)|order]] of the [[multiplicative group of integers modulo n|multiplicative group of integers modulo ''n'']].

The [[RSA (algorithm)|RSA cryptosystem]] is based on this theorem: it implies that the [[inverse function|inverse]] of the function <math>a\mapsto a^d \mod n</math> is the function <math>b\mapsto b^e \mod n,</math> where <math> e</math> is the [[multiplicative inverse]] of <math> d</math> modulo <math>\varphi(n)</math>. The difficulty of decoding without knowing the secret key, is thus the difficulty of computing <math> e,</math> which is the same as factoring <math>n.</math>

==Other formulae involving φ==

*<math>a\mid b</math> implies <math>\varphi(a)\mid\varphi(b).</math>

*<math> n \mid \varphi(a^n-1)</math> &nbsp; &nbsp; &nbsp; (''a'', ''n'' > 1)

*<math>\varphi(mn) = \varphi(m)\varphi(n)\cdot\frac{d}{\varphi(d)}</math>  &nbsp; &nbsp; &nbsp; where ''d'' = gcd(''m'', ''n''). Note the special cases
*<math>
\varphi(2m) = 
\begin{cases}
2\varphi(m) &\text{ if } m \text{ is even} \\
\varphi(m)   &\text{ if } m \text{ is odd}
\end{cases}
</math>
and
*<math>\;\varphi\left(n^m\right) = n^{m-1}\varphi(n).
</math>

*<math>\varphi(\mathrm{lcm}(m,n))\cdot\varphi(\mathrm{gcd}(m,n)) = \varphi(m)\cdot\varphi(n).</math>  &nbsp; &nbsp; &nbsp;

:::Compare this to the formula &nbsp; &nbsp; &nbsp; <math>\mathrm{lcm}(m,n)\cdot \mathrm{gcd}(m,n) = m \cdot n.</math> &nbsp; &nbsp; &nbsp; (See [[Least common multiple|lcm]]).

*<math>\varphi(n)\;</math> is even for <math>n \geq 3.</math> Moreover, if ''n'' has ''r'' distinct odd prime factors, <math>2^r \mid \varphi(n).</math>

* For any ''a'' > 1 and ''n'' > 6 such that <math> 4 \nmid n </math> there exists an <math> l \geq 2n </math> such that <math> l \mid \varphi(a^n-1)</math>.

*<math>\sum_{d \mid n} \frac{\mu^2(d)}{\varphi(d)} = \frac{n}{\varphi(n)}</math> &nbsp; &nbsp; &nbsp;<ref>Dineva (in external refs), prop. 1</ref>

*<math>\sum_{1\le k\le n \atop (k,n)=1}\!\!k = \frac{1}{2}n\varphi(n)\text{ for }n>1</math>

*<math>\sum_{k=1}^n\varphi(k) = \frac{1}{2}\left(1+ \sum_{k=1}^n \mu(k)\left\lfloor\frac{n}{k}\right\rfloor^2\right)</math>

*<math>\sum_{k=1}^n\frac{\varphi(k)}{k} = \sum_{k=1}^n\frac{\mu(k)}{k}\left\lfloor\frac{n}{k}\right\rfloor=\frac6{\pi^2}n+\mathcal{O}\left((\log n)^{2/3}(\log\log n)^{4/3}\right)</math>&nbsp;<ref name=Wal1963>{{cite book | zbl=0146.06003 | last=Walfisz | first=Arnold | authorlink=Arnold Walfisz | title=Weylsche Exponentialsummen in der neueren Zahlentheorie | language=German | series=Mathematische Forschungsberichte | volume=16 | location=Berlin | publisher=VEB Deutscher Verlag der Wissenschaften | year=1963 }}</ref>

*<math>\sum_{k=1}^n\frac{k}{\varphi(k)} = \frac{315\zeta(3)}{2\pi^4}n-\frac{\log n}2+\mathcal{O}\left((\log n)^{2/3}\right)</math>&nbsp;<ref>R. Sitaramachandrarao. On an error term of Landau II, Rocky Mountain J. Math. 15 (1985), 579-588</ref>

*<math>\sum_{k=1}^n\frac{1}{\varphi(k)} = \frac{315\zeta(3)}{2\pi^4}\left(\log n+\gamma-\sum_{p\text{ prime}}\frac{\log p}{p^2-p+1}\right)+\mathcal{O}\left(\frac{(\log n)^{2/3}}n\right)</math>&nbsp;<ref> Also R. Sitaramachandrarao (loc. cit.)</ref>

(here ''γ'' is the Euler constant).

*<math>\sum_{1\le k\le n \atop (k,m)=1} 1 = n \frac {\varphi(m)}{m} + 
\mathcal{O} \left ( 2^{\omega(m)} \right ),</math>

where ''m'' > 1 is a positive integer and ω(''m'') is the number of distinct prime factors of ''m''. (''a'', ''b'') is a standard abbreviation for gcd(''a'', ''b'').<ref>Bordellès in the [[#External links|external links]]</ref>

===Menon's identity===

{{Main|Arithmetic_function#Menon.27s_identity|l1=Menon's identity}}
In 1965 P. Kesava Menon proved
:<math>
\sum_{\stackrel{1\le k\le n}{ \gcd(k,n)=1}} \gcd(k-1,n)
=\varphi(n)d(n),
</math>
where [[Divisor function|''d''(''n'') = &sigma;<sub>0</sub>(''n'')]] is the number of divisors of ''n''.

===Formulae involving the golden ratio===

Schneider<ref>All formulae in the section are from Schneider (in the external links)</ref> found a pair of identities connecting the totient function, the [[golden ratio]] and the [[Möbius function]] <math>\mu(n)</math>. In this section <math>\varphi(n)</math> is the totient function, and <math>
\phi = \frac{1+\sqrt{5}}{2}= 1.618\dots
</math>
is the golden ratio.

They are:
:<math>
\phi=-\sum_{k=1}^\infty\frac{\varphi(k)}{k}\log\left(1-\frac{1}{\phi^k}\right)
</math>
and
:<math>
\frac{1}{\phi}=-\sum_{k=1}^\infty\frac{\mu(k)}{k}\log\left(1-\frac{1}{\phi^k}\right).
</math>
Subtracting them gives
:<math>
\sum_{k=1}^\infty\frac{\mu(k)-\varphi(k)}{k}\log\left(1-\frac{1}{\phi^k}\right)=1.
</math>
Applying the exponential function to both sides of the preceding identity yields an infinite product formula for [[Euler's constant]] {{mvar|e}}

:<math>e= \prod_{k=1}^{\infty} \left(1-\frac{1}{\phi^k}\right)^\frac{\mu(k)-\varphi(k)}{k}. </math>

The proof is based on the formulae
:<math>
\sum_{k=1}^\infty\frac{\varphi(k)}{k}(-\log(1-x^k))=\frac{x}{1-x}
</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>
\sum_{k=1}^\infty\frac{\mu(k)}{k}(-\log(1-x^k))=x, 
</math> &nbsp; &nbsp;  valid for 0 < ''x'' < 1.

==Generating functions==

The [[Dirichlet series]] for φ(''n'') may be written in terms of the [[Riemann zeta function]] as:<ref>Hardy & Wright, thm. 288</ref>
:<math>\sum_{n=1}^\infty \frac{\varphi(n)}{n^s}=\frac{\zeta(s-1)}{\zeta(s)}.</math>

The [[Lambert series]] generating function is<ref>Hardy & Wright, thm. 309</ref>

:<math>\sum_{n=1}^{\infty} \frac{\varphi(n) q^n}{1-q^n}= \frac{q}{(1-q)^2}</math>

which converges for |''q''| < 1.

Both of these are proved by elementary series manipulations and the formulae for φ(''n'').

==Growth of the function==

In the words of Hardy & Wright, φ(''n'') is "always &lsquo;nearly ''n''&rsquo;."<ref>Hardy & Wright, intro to § 18.4</ref>

First<ref>Hardy & Wright, thm. 326</ref>
:<math>
\lim\sup \frac{\varphi(n)}{n}= 1,
</math>
but as ''n'' goes to infinity,<ref>Hardy & Wright, thm. 327</ref> for all δ > 0
:<math>
\frac{\varphi(n)}{n^{1-\delta}}\rightarrow\infty.
</math>
These two formulae can be proved by using little more than the formulae for φ(''n'') and the [[divisor function|divisor sum function]] σ(''n'').<br>
In fact, during the proof of the second formula, the inequality
:<math>
\frac {6}{\pi^2} < \frac{\varphi(n) \sigma(n)}{n^2} < 1,
</math>
true for ''n'' > 1, is proven.<br> 
We also have<ref>Hardy & Wright, thm. 328</ref>
:<math>
\lim\inf\frac{\varphi(n)}{n}\log\log n = e^{-\gamma}.
</math>
Here &gamma; is [[Euler–Mascheroni constant|Euler's constant]], &nbsp; &gamma; =  0.577215665..., &nbsp; e<sup>γ</sup> = 1.7810724..., &nbsp; e<sup>−γ</sup> = 0.56145948... .

Proving this doesn't quite require the [[prime number theorem]].<ref>In fact Chebychev's theorem (Hardy & Wright, thm.7) and
Mertens' third theorem is all that's needed</ref><ref>Hardy & Wright, thm. 436</ref> Since log log (''n'') goes to infinity, this formula shows that
:<math>
\lim\inf\frac{\varphi(n)}{n}= 0.
</math>

In fact, more is true.<ref>Bach & Shallit, thm. 8.8.7</ref><ref name=Rib320>Ribenboim, p.320</ref>

:<math>
\varphi(n) > \frac {n} {e^\gamma\; \log \log n + \frac {3} {\log \log n}} 
</math> &nbsp;  &nbsp;  &nbsp; for ''n'' > 2, and

:<math>
\varphi(n) < \frac  {n} {e^{ \gamma}\log \log n} 
</math> &nbsp;  &nbsp;  &nbsp;  &nbsp;  &nbsp;  &nbsp;  &nbsp;  &nbsp;  &nbsp; &nbsp;  &nbsp; for infinitely many ''n''.

Concerning the second inequality, Ribenboim says "The method of proof is interesting, in that the inequality is shown first under the assumption that the [[Riemann hypothesis]] is true, secondly under the contrary assumption."<ref name=Rib320/>

For the average order, we have<ref name=Wal1963/><ref name=SMC2425>Sándor, Mitrinović & Crstici (2006) pp.24-25</ref>
:<math>
\varphi(1)+\varphi(2)+\cdots+\varphi(n) = \frac{3n^2}{\pi^2}+\mathcal{O}\left(n(\log n)^{2/3}(\log\log n)^{4/3}\right)\ \ (n\rightarrow\infty),
</math>
due to [[Arnold Walfisz]], its proof exploiting estimates on exponential sums due to  [[Ivan Matveevich Vinogradov|I. M. Vinogradov]] and [[N.M. Korobov]] (this is currently the best known estimate of this type). The [[Big O notation|"Big O"]] stands for a quantity that is bounded by a constant times the function of "n" inside the parentheses (which is small compared to ''n''<sup>2</sup>).

This result can be used to prove<ref>Hardy & Wright, thm. 332</ref>  that the probability of two randomly chosen numbers being relatively prime is &nbsp;<math>\tfrac{6}{\pi^2}.</math>

==Ratio of consecutive values==

In 1950 Somayajulu proved<ref name=Rib38>Ribenboim, p.38</ref><ref name=SMC16>Sándor, Mitrinović & Crstici (2006) p.16</ref>
:<math>
\lim\inf \frac{\varphi(n+1)}{\varphi(n)}= 0
</math> &nbsp;  &nbsp;  &nbsp;  &nbsp; and  &nbsp;  &nbsp;  &nbsp;  &nbsp; 
:<math>
\lim\sup \frac{\varphi(n+1)}{\varphi(n)}= \infty.
</math>
In 1954 [[Andrzej Schinzel|Schinzel]] and [[Wacław Sierpiński|Sierpiński]] strengthened this, proving<ref name=Rib38/><ref name=SMC16/> that the set 
:<math>
\left\{\frac{\varphi(n+1)}{\varphi(n)},\;\;n = 1,2,\cdots\right\}
</math>
is [[Dense set|dense]] in the positive real numbers.
They also proved<ref name=Rib38/> that the set 
:<math>
\left\{\frac{\varphi(n)}{n},\;\;n = 1,2,\cdots\right\}
</math>
is dense in the interval (0, 1).

==Totient numbers==
A '''totient number''' is a value of Euler's totient function: that is, an ''m'' for which there is at least one ''x'' for which φ(''x'')&nbsp;=&nbsp;''m''.  The ''valency'' or ''multiplicity'' of a totient number ''m'' is the number of solutions to this equation.<ref name=Guy144>Guy (2004) p.144</ref>   A ''[[nontotient]]'' is a natural number which is not a totient number: there are infinitely many nontotients, <ref name=SC230>Sándor & Crstici (2004) p.230</ref> and indeed every odd number has a multiple which is a nontotient.<ref name=Zha1993>{{cite journal | zbl=0772.11001 | last=Zhang | first=Mingzhi | title=On nontotients | journal=[[Journal of Number Theory]] | volume=43 | number=2 | pages=168–172 | year=1993 | issn=0022-314X }}</ref>

The number of totient numbers up to a given limit ''x'' is
:<math>\frac{x}{\log x}\exp\left({ (C+o(1))(\log\log\log x)^2 }\right) \ </math>
for a constant ''C'' = 0.8178146... .<ref name=Ford1998>{{cite journal | zbl=0914.11053 | last=Ford | first=Kevin | title=The distribution of totients | journal=Ramanujan J. | volume=2 | number=1-2 | pages=67–151 | year=1998 | issn=1382-4090 }}</ref>

If counted accordingly to multiplicity, the number of totient numbers up to a given limit ''x'' is 
:<math>\vert\{ n : \phi(n) \le x \}\vert = \frac{\zeta(2)\zeta(3)}{\zeta(6)} \cdot x + R(x) \ </math>
where the error term ''R'' is of order at most <math>x / (\log x)^k</math> for any positive ''k''.<ref name=SMC22>Sándor et al (2006) p.22</ref>  

It is known that the multiplicity of ''m'' exceeds ''m''<sup>δ</sup> infinitely often for any δ < 0.55655.<ref name=SMC21>Sándor et al (2006) p.21</ref><ref name=Guy145>Guy (2004) p.145</ref>


===Ford's theorem===

{{harvtxt|Ford|1999}} proved that for every integer ''k''&nbsp;≥&nbsp;2 there is a totient number ''m'' of multiplicity ''k'': that is, for which the equation φ(''x'')&nbsp;=&nbsp;''m'' has exactly ''k'' solutions; this result had previously been conjectured by [[Wacław Sierpiński]],<ref name=SC229>Sándor & Crstici (2004) p.229</ref> and it had been obtained as a consequence of [[Schinzel's hypothesis H]].<ref name=Ford1998/>  Indeed, each multiplicity that occurs, does so infinitely often.<ref name=Ford1998/><ref name=Guy145/>

However, no number ''m'' is known with multiplicity ''k''&nbsp;=&nbsp;1. [[Carmichael's totient function conjecture]] is the statement that there is no such ''m''.<ref name=SC228>Sándor & Crstici (2004) p.228</ref>

==Applications==

===Cyclotomy===

{{main|Constructible polygon}}

In the last section of the ''Disquisitiones''<ref>Gauss, DA. The 7th § is arts. 336-366</ref><ref>Gauss proved if ''n'' satisfies certain conditions then the ''n''-gon can be constructed. In 1837 [[Pierre Wantzel]] proved the converse, if the ''n''-gon is constructible, then ''n'' must satisfy Gauss's conditions</ref> Gauss proves<ref>Gauss, DA, art 366</ref> that a regular ''n''-gon can be constructed with straightedge and compass if φ(''n'') is a power of 2. If ''n'' is a  power of an odd prime number the formula for the totient says its totient can be a power of two only if a) ''n'' is a first power and b) ''n''&nbsp;−&nbsp;1 is a power of 2. The primes that are one more than a power of 2 are called [[Fermat prime]]s, and only five are known: 3, 5, 17, 257, and 65537. Fermat and Gauss knew of these. Nobody has been able to prove whether there are any more.

Thus, a regular ''n''-gon has a straightedge-and-compass construction if ''n'' is a product of distinct Fermat primes and any power of 2.<br>
The first few such ''n'' are<ref>Gauss, DA, art. 366. This list is the last sentence in the ''Disquisitiones''</ref> 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, ... . &nbsp; &nbsp; {{OEIS|A003401}}

===The RSA cryptosystem===

{{main|RSA (algorithm)}}

Setting up an RSA system involves choosing large prime numbers ''p'' and ''q'', computing ''n'' = ''pq'' and ''k'' = φ(''n''), and finding two numbers ''e'' and ''d'' such that ''ed'' ≡ 1 (mod ''k''). The numbers ''n'' and ''e'' (the "encryption key") are released to the public, and ''d'' (the "decryption key") is kept secure.

A message, represented by an integer ''m'', where 0 < ''m'' < ''n'', is encrypted by computing ''S'' = ''m''<sup>''e''</sup> (mod ''n'').

It is decrypted by computing ''t'' = ''S''<sup>''d''</sup> (mod ''n''). Euler's Theorem can be used to show that if 0 < ''t'' < ''n'', then ''t'' = ''m''. 

The security of an RSA system would be compromised if the number ''n'' could be factored or if φ(''n'') could be computed without factoring ''n''.

==Unsolved problems==

===Lehmer's conjecture===

{{main|Lehmer's totient problem}}

If ''p'' is prime, then φ(''p'') = ''p'' − 1. In 1932 [[D. H. Lehmer]] asked if there are any composite numbers ''n'' such that φ(''n'') | ''n'' − 1. None are known.<ref>Ribenboim, pp. 36-37.</ref>

In 1933 he proved that if any such ''n'' exists, it must be odd, square-free, and divisible by at least seven primes (i.e.  ω(''n'') ≥  7). In 1980 Cohen and Hagis proved that ''n'' > 10<sup>20</sup> and that ω(''n'') ≥  14.<ref>{{cite journal | zbl=0436.10002 | last1=Cohen | first1=Graeme L. | last2=Hagis | first2=Peter, jun. | title=On the number of prime factors of ''n'' if φ(''n'') divides ''n''−1 | journal=Nieuw Arch. Wiskd., III. Ser. | volume=28 | pages=177–185 | year=1980 | issn=0028-9825 }}</ref> Further, Hagis showed that if 3 divides ''n'' then ''n'' > 10<sup>1937042</sup> and ω(''n'') ≥  298848.<ref>{{cite journal | zbl=0668.10006 | last=Hagis | first=Peter, jun. | title=On the equation ''M''⋅φ(''n'')=''n''−1 | journal=Nieuw Arch. Wiskd., IV. Ser. | volume=6 | number=3 | pages=255–261 | year=1988 | issn=0028-9825 }}</ref><ref name=Guy142>Guy (2004) p.142</ref>

===Carmichael's conjecture===

{{main|Carmichael's totient function conjecture}}

This states that there is no number ''n'' with the property that for all other numbers ''m'', ''m'' ≠ ''n'', φ(''m'') ≠ φ(''n''). See [[#Ford's theorem|Ford's theorem]] above.

As stated in the main article, if there is a single counterexample to this conjecture, there must be infinitely many counterexamples, and the smallest one has at least ten billion digits in base 10.<ref name=Guy144/>

== See also ==
*[[Carmichael function]]
*[[Duffin–Schaeffer conjecture]]
*[[Fermat's little theorem#Generalizations|Generalizations of Fermat's little theorem]]
*[[Highly composite number]]
*[[Multiplicative group of integers modulo n]]
*[[Ramanujan sum]]

== Notes ==
{{reflist|colwidth=30em}}

==References==
{{refbegin|colwidth=30em}}

The ''[[Disquisitiones Arithmeticae]]'' has been translated from Latin into English and German. The German edition includes all of Gauss' papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. 

References to the ''Disquisitiones'' are of the form Gauss, DA, art. ''nnn''.

*{{citation
 | last1 = Abramowitz | first1 = M. | author1-link = Milton Abramowitz
 | last2 = Stegun | first2 = I. A. | author2-link = Irene A. Stegun
 | isbn = 0-486-61272-4
 | location = New York
 | publisher = [[Dover Publications]]
 | title = Handbook of Mathematical Functions
 | year = 1964}}.  See paragraph 24.3.2.
*{{citation
  | last1 = Bach  | first1 = Eric | author1-link = Eric Bach
  | last2 = Shallit | first2 = Jeffrey | author2-link = Jeffrey Shallit
  | title = Algorithmic Number Theory (Vol I: Efficient Algorithms)
  | publisher = [[The MIT Press]]
  | location = Cambridge, MA
  | year = 1996
  | isbn = 0-262-02405-5
  | zbl=0873.11070
  | series=MIT Press Series in the Foundations of Computing
}}
*{{citation
 | last = Ford | first = Kevin
 | doi = 10.2307/121103
 | mr = 1715326 | zbl=0978.11053
 | issn= 0003-486X
 | issue = 1
 | journal = [[Annals of Mathematics]]
 | jstor = 121103
 | pages = 283–311
 | title = The number of solutions of φ(''x'')&nbsp;=&nbsp;''m''
 | volume = 150
 | year = 1999}}.
*{{citation
  | last1 = Gauss  | first1 = Carl Friedrich | author1-link = Carl Friedrich Gauss
  | last2 = Clarke | first2 = Arthur A. (translator into English)  
  | title = Disquisitiones Arithemeticae (Second, corrected edition)
  | publisher = [[Springer Publishing|Springer]]
  | location = New York
  | year = 1986
  | isbn = 0-387-96254-9}}
*{{citation
  | last1 = Gauss  | first1 = Carl Friedrich | author1-link = Carl Friedrich Gauss
  | last2 = Maser | first2 = H. (translator into German)  
  | title = Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)
  | publisher = Chelsea
  | location = New York
  | year = 1965
  | isbn = 0-8284-0191-8}}
*{{citation
  | last1 = Graham  | first1 = Ronald | author1-link = Ronald Graham
  | last2 = Knuth  | first2 = Donald | author2-link = Donald Knuth
  | last3 = Patashnik  | first3 = Oren | author3-link = Oren Patashnik
  | title = [[Concrete Mathematics]]: a foundation for computer science
  | edition=2nd
  | publisher = Addison-Wesley
  | location = Reading, MA
  | year = 1994
  | isbn = 0-201-55802-5 | zbl=0836.00001}}
* {{citation | first=Richard K. | last=Guy | authorlink=Richard K. Guy | title=[[Unsolved Problems in Number Theory]] | edition=3rd | publisher=[[Springer-Verlag]] | year=2004 | isbn=0-387-20860-7 | zbl=1058.11001 | series=Problem Books in Mathematics | location=New York, NY }}
*{{citation
  | last1 = Hardy  | first1 = G. H. | author1-link = G. H. Hardy
  | last2 = Wright | first2 = E. M. | author2-link = E. M. Wright
  | title = An Introduction to the Theory of Numbers (Fifth edition)
  | publisher = [[Oxford University Press]]
  | location = Oxford
  | year = 1980
  | isbn = 978-0-19-853171-5}}
* {{citation | first1 = Calvin T. | last1 = Long | year = 1972 | title = Elementary Introduction to Number Theory | edition = 2nd | publisher = [[D. C. Heath and Company]] | location = Lexington | lccn = 77-171950 }}
* {{citation | first1 = Anthony J. | last1 = Pettofrezzo | first2 = Donald R. | last2 = Byrkit | year = 1970 | title = Elements of Number Theory | publisher = [[Prentice Hall]] | location = Englewood Cliffs | lccn = 77-81766 }}
*{{citation
  | last1 = Ribenboim  | first1 = Paulo | authorlink = Paulo Ribenboim
  | title = The New Book of Prime Number Records | edition=3rd
  | publisher = [[Springer Science+Business Media|Springer]]
  | location = New York
  | year = 1996
  | zbl=0856.11001
  | isbn = 0-387-94457-5}}
*{{citation
  | last1 = Sandifer  | first1 = Charles
  | title = The early mathematics of Leonhard Euler
  | publisher = MAA
  | year = 2007
  | isbn = 0-88385-559-3}}
* {{citation | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=[[Springer-Verlag]] | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 | pages=9–36}}
* {{cite book | last1=Sándor | first1=Jozsef | last2=Crstici | first2=Borislav | title=Handbook of number theory II | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | zbl=1079.11001 | pages=179–327 }}
*{{citation
 | last = Schramm | first = Wolfgang
 | issue = 8(1)
 | journal = Electronic Journal of Combinatorial Number Theory
 | title = The Fourier transform of functions of the greatest common divisor
 | volume = A50
 | year = 2008
 |url=http://www.integers-ejcnt.org/vol8.html }}.
{{refend}}

==External links==<!-- This section is linked from [[Euler's totient function]] -->
* {{springer|title=Totient function|id=p/t110040}}
*Kirby Urner, ''[http://groups.google.com/group/k12.ed.math/browse_thread/thread/19f74d278e88b65d/bd50b5ae25c74465 Computing totient function in Python and scheme]'', (2003)
*[http://www.javascripter.net/math/calculators/eulertotientfunction.htm Euler's totient function calculator in JavaScript &mdash; up to 20 digits]
* Bordellès, Olivier, [http://www.les-maths.net/phorum/read.php?5,359275,359275 Numbers prime to ''q'' in <math>[1, n]</math>]
*Dineva, Rosica, [http://www.mtholyoke.edu/~robinson/reu/reu05/rdineva1.pdf The Euler Totient, the Möbius, and the Divisor Functions]
*Miyata, Daisuke & Yamashita, Michinori, [http://yamashita-lab.net/open/mathconf-0.pdf Derived logarithmic function of Euler's function]
*Plytage, Loomis, Polhill [http://facstaff.bloomu.edu/jpolhill/cmj034-042.pdf Summing Up The Euler Phi Function]
*Schneider, Robert P. [http://arxiv.org/ftp/arxiv/papers/1211/1211.2025.pdf A Golden Product Identity for {{mvar|e}}.]

{{Totient}}

[[Category:Number theory]]
[[Category:Modular arithmetic]]
[[Category:Multiplicative functions]]
[[Category:Articles containing proofs]]