View source for Euler's totient function

{{Short description|Number of integers coprime to and less than n}}
{{hatnote group|
{{Redirect|φ(n)||Phi}}
{{distinguish|Euler function}}
}}
{{log(x)}}
[[Image:EulerPhi.svg|thumb|The first thousand values of {{math|''φ''(''n'')}}. The points on the top line represent {{Math|''φ''(''p'')}} when {{mvar|p}} is a prime number, which is {{Math|''p'' − 1.}}<ref>{{Cite web
| url = https://www.khanacademy.org/computing/computer-science/cryptography/modern-crypt/v/euler-s-totient-function-phi-function
| title = Euler's totient function
| website = Khan Academy
| access-date = 2016-02-26
}}</ref>]]

In [[number theory]], '''Euler's totient function''' counts the positive integers up to a given integer <math>n</math> that are [[relatively prime]] to <math>n</math>. It is written using the Greek letter [[phi]] as <math>\varphi(n)</math> or <math>\phi(n)</math>, and may also be called '''Euler's phi function'''. In other words, it is the number of integers <math>k</math> in the range <math>1\leq k\leq n</math> for which the [[greatest common divisor]] <math>\gcd(n,k)</math> is equal to 1.<ref>{{harvtxt|Long|1972|p=85}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=72}}</ref> The integers <math>k</math> of this form are sometimes referred to as [[totative]]s of <math>n</math>.

For example, the totatives of <math>n=9</math> are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since <math>\gcd(9,3)=\gcd(9,6)=3</math> and <math>\gcd(9,9)=9</math>. Therefore, <math>\varphi(9)=6</math>. As another example, <math>\varphi(1)=1</math> since for <math>n=1</math> the only integer in the range from 1 to <math>n</math> is 1 itself, and <math>\gcd(1,1)=1</math>.

Euler's totient function is a [[multiplicative function]], meaning that if two numbers <math>m</math> and <math>n</math> are relatively prime, then <math>\varphi(mn)=\varphi(m)\varphi(n)</math>.<ref>{{harvtxt|Long|1972|p=162}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=80}}</ref>
This function gives the [[order (group theory)|order]] of the [[multiplicative group of integers modulo n|multiplicative group of integers modulo {{mvar|n}}]] (the [[Multiplicative group of integers modulo n|group]] of [[unit (ring theory)|unit]]s of the [[ring (algebra)|ring]] <math>\Z/n\Z</math>).<ref>See [[#Euler's theorem|Euler's theorem]].</ref> It is also used for defining the [[RSA (cryptosystem)|RSA encryption system]].

== History, terminology, and notation ==

[[Leonhard Euler]] introduced the function in 1763.<ref>L. Euler "[http://eulerarchive.maa.org/pages/E271.html Theoremata arithmetica nova methodo demonstrata]" (An arithmetic theorem proved by a new method), ''Novi commentarii academiae scientiarum imperialis Petropolitanae'' (New Memoirs of the Saint-Petersburg Imperial Academy of Sciences), '''8''' (1763), 74–104. (The work was presented at the Saint-Petersburg Academy on October 15, 1759. A work with the same title was presented at the Berlin Academy on June 8, 1758). Available on-line in: [[Ferdinand Rudio]], {{abbr|ed.|editor}}, ''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 <math>n</math> as the number of integers that are smaller than <math>N</math> and relatively prime to <math>N</math> (... aequalis sit multitudini numerorum ipso N minorum, qui simul ad eum sint primi, ...), which is the phi function, φ(N).</ref><ref name="Sandifer, p. 203">Sandifer, p. 203</ref><ref>Graham et al. p. 133 note 111</ref> However, he did not at that time choose any specific symbol to denote it. In a 1784 publication, Euler studied the function further, choosing the Greek letter <math>\pi</math> to denote it: he wrote <math>\pi D</math> for "the multitude of numbers less than <math>D</math>, and which have no common divisor with it".<ref>L. Euler, ''[http://math.dartmouth.edu/~euler/docs/originals/E564.pdf Speculationes circa quasdam insignes proprietates numerorum]'', Acta Academiae Scientarum Imperialis Petropolitinae, vol. 4, (1784), pp. 18–30, or Opera Omnia, Series 1, volume 4, pp. 105–115. (The work was presented at the Saint-Petersburg Academy on October 9, 1775).</ref> This definition varies from the current definition for the totient function at <math>D=1</math> but is otherwise the same. The now-standard notation<ref name="Sandifer, p. 203"/><ref>Both {{math|''φ''(''n'')}} and {{math|''ϕ''(''n'')}} are seen in the literature. These are two forms of the lower-case Greek letter [[phi]].</ref> <math>\varphi(A)</math> comes from [[Gauss]]'s 1801 treatise ''[[Disquisitiones Arithmeticae]]'',<ref>Gauss, ''Disquisitiones Arithmeticae'' article&nbsp;38</ref><ref>{{cite book |last=Cajori |first=Florian |author-link=Florian Cajori |title=A History Of Mathematical Notations Volume II |year=1929 |publisher=Open Court Publishing Company|at=§409}}</ref> although Gauss did not use parentheses around the argument and wrote <math>\varphi A</math>. Thus, it is often 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 [https://books.google.com/books?id=-AcPAAAAIAAJ&pg=PA361 page 361].</ref><ref>{{cite OED2|totient}}</ref> so it is also referred to as '''Euler's totient function''', the '''Euler totient''', or '''Euler's totient'''.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Totient Function |url=https://mathworld.wolfram.com/TotientFunction.html |access-date=2025-02-09 |website=mathworld.wolfram.com |language=en}}</ref> [[Jordan's totient function|Jordan's totient]] is a generalization of Euler's.

The '''cototient''' of <math>n</math> is defined as <math>n-\varphi(n)</math>. It counts the number of positive integers less than or equal to <math>n</math> that have at least one [[prime number|prime factor]] in common with <math>n</math>.

== Computing Euler's totient function ==

There are several formulae for computing <math>\varphi(n)</math>.

===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 {{mvar|n}}.

An equivalent formulation is
:<math>\varphi(n) = p_1^{k_1-1}(p_1{-}1)\,p_2^{k_2-1}(p_2{-}1)\cdots p_r^{k_r-1}(p_r{-}1),</math>
where <math>n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}</math> is the [[prime factorization]] of <math>n</math> (that is, <math>p_1, p_2,\ldots, p_r</math> are distinct prime numbers).

The proof of these formulae depends on two important facts.

==== Phi is a multiplicative function ====

This means that if {{math|1=gcd(''m'', ''n'') = 1}}, then {{math|1=''φ''(''m'') ''φ''(''n'') = ''φ''(''mn'')}}. ''Proof outline:'' Let {{mvar|A}}, {{mvar|B}}, {{mvar|C}} be the sets of positive integers which are [[coprime]] to and less than {{mvar|m}}, {{mvar|n}}, {{mvar|mn}}, respectively, so that {{math|1={{!}}''A''{{!}} = ''φ''(''m'')}}, etc. Then there is a [[bijection]] between {{math|''A'' × ''B''}} and {{mvar|C}} by the [[Chinese remainder theorem]].

==== Value of phi for a prime power argument ====

If {{mvar|p}} is prime and {{math|''k'' ≥ 1}}, then

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

''Proof'': Since {{mvar|p}} is a prime number, the only possible values of {{math|gcd(''p''<sup>''k''</sup>, ''m'')}} are {{math|1, ''p'', ''p''<sup>2</sup>, ..., ''p''<sup>''k''</sup>}}, and the only way to have {{math|gcd(''p''<sup>''k''</sup>, ''m'') > 1}} is if {{mvar|m}} is a multiple of {{mvar|p}}, that is, {{math|1=''m'' ∈ {{mset|1=''p'', 2''p'', 3''p'', ..., ''p''<sup>''k'' − 1</sup>''p'' = ''p''<sup>''k''</sup>}}}}, and there are {{math|''p''<sup>''k'' − 1</sup>}} such multiples not greater than {{math|''p''<sup>''k''</sup>}}. Therefore, the other {{math|''p''<sup>''k''</sup> − ''p''<sup>''k'' − 1</sup>}} numbers are all relatively prime to {{math|''p''<sup>''k''</sup>}}.

====Proof of Euler's product formula====

The [[fundamental theorem of arithmetic]] states that if {{math|''n'' > 1}} there is a unique expression <math>n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}, </math> where {{math|''p''<sub>1</sub> < ''p''<sub>2</sub> < ... < ''p''<sub>''r''</sub>}} are [[prime number]]s and each {{math|''k''<sub>''i''</sub> ≥ 1}}. (The case {{math|1=''n'' = 1}} corresponds to the [[empty product]].) Repeatedly using the multiplicative property of {{mvar|φ}} and the formula for {{math|''φ''(''p''<sup>''k''</sup>)}} gives

:<math>\begin{array} {rcl}
\varphi(n)&=& \varphi(p_1^{k_1})\, \varphi(p_2^{k_2}) 
\cdots\varphi(p_r^{k_r})\\[.1em]
&=& 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)\\[.1em]
&=& 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)\\[.1em]
&=&n \left(1- \frac{1}{p_1} \right)\left(1- \frac{1}{p_2} \right) \cdots\left(1- \frac{1}{p_r} \right).
\end{array}</math>

This gives both versions of Euler's product formula.

An alternative proof that does not require the multiplicative property instead uses the [[inclusion-exclusion principle]] applied to the set <math>\{1,2,\ldots,n\}</math>, excluding the sets of integers divisible by the prime divisors.

====Example====

:<math>\varphi(20)=\varphi(2^2 5)=20\,(1-\tfrac12)\,(1-\tfrac15)
=20\cdot\tfrac12\cdot\tfrac45=8.</math>

In words: the distinct prime factors of 20 are 2 and 5; half of the twenty integers from 1 to 20 are divisible by 2, leaving ten; a fifth of those are divisible by 5, leaving eight numbers coprime to 20; these are: 1, 3, 7, 9, 11, 13, 17, 19.

The alternative formula uses only integers:<math display="block">\varphi(20) = \varphi(2^2 5^1)= 2^{2-1}(2{-}1)\,5^{1-1}(5{-}1) = 2\cdot 1\cdot 1\cdot 4 = 8.</math>

===Fourier transform===

The totient is the [[discrete Fourier transform]] of the [[Greatest common divisor|gcd]], evaluated at 1.<ref>{{harvtxt|Schramm|2008}}</ref> Let

:<math> \mathcal{F} \{ \mathbf{x} \}[m] = \sum\limits_{k=1}^n x_k \cdot e^{{-2\pi i}\frac{mk}{n}}</math>

where {{math|''x<sub>k</sub>'' {{=}} gcd(''k'',''n'')}} for {{math|''k'' ∈ {1, ..., ''n''}<nowiki/>}}. Then

:<math>\varphi (n) = \mathcal{F} \{ \mathbf{x} \}[1] = \sum\limits_{k=1}^n \gcd(k,n) e^{-2\pi i\frac{k}{n}}.</math>

The real part of this formula is

:<math>\varphi (n)=\sum\limits_{k=1}^n \gcd(k,n) \cos {\tfrac{2\pi k}{n}}
.</math>

For example, using <math>\cos\tfrac{\pi}5 = \tfrac{\sqrt 5+1}4
</math> and <math>\cos\tfrac{2\pi}5 = \tfrac{\sqrt 5-1}4
</math>:<math display="block">\begin{array}{rcl}
\varphi(10) &=& \gcd(1,10)\cos\tfrac{2\pi}{10} + \gcd(2,10)\cos\tfrac{4\pi}{10}
+ \gcd(3,10)\cos\tfrac{6\pi}{10}+\cdots+\gcd(10,10)\cos\tfrac{20\pi}{10}\\
&=& 1\cdot(\tfrac{\sqrt5+1}4)  + 2\cdot(\tfrac{\sqrt5-1}4) + 
1\cdot(-\tfrac{\sqrt5-1}4) + 2\cdot(-\tfrac{\sqrt5+1}4) + 5\cdot (-1) \\
&& +\  2\cdot(-\tfrac{\sqrt5+1}4) + 1\cdot(-\tfrac{\sqrt5-1}4) + 2\cdot(\tfrac{\sqrt5-1}4) +
1\cdot(\tfrac{\sqrt5+1}4) + 10 \cdot (1) \\
&=& 4 .
\end{array}
</math>Unlike the [[Euler product]] and the divisor sum formula, this one does not require knowing the factors of {{mvar|n}}. However, it does involve the calculation of the greatest common divisor of {{mvar|n}} and every positive integer less than {{mvar|n}}, which suffices to provide the factorization anyway.

===Divisor sum===

The property established by Gauss,<ref>Gauss, DA, art 39</ref> that

:<math>\sum_{d\mid n}\varphi(d)=n,</math>

where the sum is over all positive divisors {{mvar|d}} of {{mvar|n}}, can be proven in several ways. (See [[Arithmetical function#Notation|Arithmetical function]] for notational conventions.)

One proof is to note that {{math|''φ''(''d'')}} is also equal to the number of possible generators of the [[cyclic group]] {{math|''C''<sub>''d''</sub>}} ; specifically, if {{math|''C''<sub>''d''</sub> {{=}} ⟨''g''⟩}} with {{math|1=''g''<sup>''d''</sup> = 1}}, then {{math|''g''<sup>''k''</sup>}} is a generator for every {{mvar|k}} coprime to {{mvar|d}}. Since every element of {{math|''C''<sub>''n''</sub>}} generates a cyclic [[subgroup]], and each subgroup {{math|''C''<sub>''d''</sub> ⊆ ''C''<sub>''n''</sub>}} is generated by precisely {{math|''φ''(''d'')}} elements of {{math|''C''<sub>''n''</sub>}}, the formula follows.<ref>Gauss, DA art. 39, arts. 52-54</ref> Equivalently, the formula can be derived by the same argument applied to the [[Root of unity#Group of nth roots of unity|multiplicative group of the {{mvar|n}}th roots of unity]] and the [[primitive root of unity|primitive {{mvar|d}}th roots of unity]].

The formula can also be derived from elementary arithmetic.<ref>Graham et al. pp. 134-135</ref> For example, let {{math|''n'' {{=}} 20}} and consider the positive fractions up to 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>

These twenty fractions are all the positive {{sfrac|''k''|''d''}} ≤ 1 whose denominators are the divisors {{math|''d''  {{=}} 1, 2, 4, 5, 10, 20}}. The fractions with 20 as denominator are those with numerators relatively prime to 20, namely {{sfrac|1|20}}, {{sfrac|3|20}}, {{sfrac|7|20}}, {{sfrac|9|20}}, {{sfrac|11|20}}, {{sfrac|13|20}}, {{sfrac|17|20}}, {{sfrac|19|20}}; by definition this is {{math|''φ''(20)}} fractions. Similarly, there are {{math|''φ''(10)}} fractions with denominator 10, and {{math|''φ''(5)}} fractions with denominator 5, etc. Thus the set of twenty fractions is split into subsets of size {{math|''φ''(''d'')}} for each {{math|''d''}} dividing 20. A similar argument applies for any ''n.''

[[Möbius inversion]] applied to the divisor sum formula gives
:<math> \varphi(n) = \sum_{d\mid n} \mu\left( d \right) \cdot \frac{n}{d}  = n\sum_{d\mid n} \frac{\mu (d)}{d},</math>

where {{mvar|μ}} is the [[Möbius function]], the [[multiplicative function]] defined by <math>\mu(p) = -1</math> and <math> \mu(p^k) = 0</math> for each prime {{math|1=''p''}} and {{math|1=''k'' ≥ 2}}. This formula may also be derived from the product formula by multiplying out <math display="inline"> \prod_{p\mid n} (1 - \frac{1}{p}) </math> to get <math display="inline"> \sum_{d \mid n} \frac{\mu (d)}{d}. </math>

An example:<math display="block">
 \begin{align}
\varphi(20) &= \mu(1)\cdot 20 + \mu(2)\cdot 10 +\mu(4)\cdot 5 +\mu(5)\cdot 4 + \mu(10)\cdot 2+\mu(20)\cdot 1\\[.5em]
&= 1\cdot 20 - 1\cdot 10 + 0\cdot 5 - 1\cdot 4 + 1\cdot 2 + 0\cdot 1 = 8.
\end{align}
</math>

==Some values==

The first 100 values {{OEIS|A000010}} are shown in the table and graph below:

[[File:EulerPhi100.svg|thumb|Graph of the first 100 values]]
:{| class="wikitable" style="text-align: right"
|+{{math|''φ''(''n'')}} for {{math|1 ≤ ''n'' ≤ 100}}
! +
! 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10
|- 
! 0 
| 1 || 1 || 2 || 2 || 4 || 2 || 6 || 4 || 6 || 4 
|-
! 10
| 10 || 4 || 12 || 6 || 8 || 8 || 16 || 6 || 18 || 8 
|- 
! 20
| 12 || 10 || 22 || 8 || 20 || 12 || 18 || 12 || 28 || 8 
|-
! 30
| 30 || 16 || 20 || 16 || 24 || 12 || 36 || 18 || 24 || 16 
|-
! 40
| 40 || 12 || 42 || 20 || 24 || 22 || 46 || 16 || 42 || 20
|-
! 50
| 32 || 24 || 52 || 18 || 40 || 24 || 36 || 28 || 58 || 16 
|-
! 60
| 60 || 30 || 36 || 32 || 48 || 20 || 66 || 32 || 44 || 24 
|-
! 70
| 70 || 24 || 72 || 36 || 40 || 36 || 60 || 24 || 78 || 32
|-
! 80
| 54 || 40 || 82 || 24 || 64 || 42 || 56 || 40 || 88 || 24
|-
! 90
| 72 || 44 || 60 || 46 || 72 || 32 || 96 || 42 || 60 || 40 
|}

In the graph at right the top line {{math|''y'' {{=}} ''n'' − 1}} is an [[upper bound]] valid for all {{mvar|n}} other than one, and attained if and only if {{mvar|n}} is a prime number. A simple lower bound is <math>\varphi(n) \ge \sqrt{n/2} </math>, which is rather loose: in fact, the [[Limit superior and limit inferior|lower limit]] of the graph is proportional to {{math|{{sfrac|''n''|log log ''n''}}}}.<ref name="hw328"/>
{{clear}}

==Euler's theorem==

{{main|Euler's theorem}}

This states that if {{mvar|a}} and {{mvar|n}} are [[relatively prime]] then

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

The special case where {{mvar|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 {{math|''φ''(''n'')}} is the [[order (group theory)|order]] of the [[multiplicative group of integers modulo n|multiplicative group of integers modulo {{mvar|n}}]].

The [[RSA (algorithm)|RSA cryptosystem]] is based on this theorem: it implies that the [[inverse function|inverse]] of the function {{math|''a'' ↦ ''a<sup>e</sup>'' mod ''n''}}, where {{mvar|e}} is the (public) encryption exponent, is the function {{math|''b'' ↦ ''b<sup>d</sup>'' mod ''n''}}, where {{mvar|d}}, the (private) decryption exponent, is the [[multiplicative inverse]] of {{mvar|e}} modulo {{math|''φ''(''n'')}}. The difficulty of computing {{math|''φ''(''n'')}} without knowing the factorization of {{mvar|n}} is thus the difficulty of computing {{mvar|d}}: this is known as the [[RSA problem]] which can be solved by factoring {{mvar|n}}. The owner of the private key knows the factorization, since an RSA private key is constructed by choosing {{mvar|n}} as the product of two (randomly chosen) large primes {{mvar|p}} and {{mvar|q}}. Only {{mvar|n}} is publicly disclosed, and given the [[Integer factorization|difficulty to factor large numbers]] we have the guarantee that no one else knows the factorization.

==Other formulae==

*<math>a\mid b \implies \varphi(a)\mid\varphi(b)</math>
*<math> m \mid \varphi(a^m-1)</math>
*<math>\varphi(mn) = \varphi(m)\varphi(n)\cdot\frac{d}{\varphi(d)} \quad\text{where }d = \operatorname{gcd}(m,n)</math>
**In particular:
*<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>
*<math>\varphi\left(n^m\right) = n^{m-1}\varphi(n)</math>
*<math>\varphi(\operatorname{lcm}(m,n))\cdot\varphi(\operatorname{gcd}(m,n)) = \varphi(m)\cdot\varphi(n)</math>
::Compare this to the formula <math display=inline>\operatorname{lcm}(m,n)\cdot \operatorname{gcd}(m,n) = m \cdot n</math>  (see [[least common multiple]]).
*{{math|''φ''(''n'')}} is even for {{math|''n'' ≥ 3}}. <br>Moreover, if {{mvar|n}} has {{mvar|r}} distinct odd prime factors, {{math|2<sup>''r''</sup> {{!}} ''φ''(''n'')}}
*For any {{math|''a'' > 1}} and {{math|''n'' > 6}} such that {{math|4 ∤ ''n''}} there exists an {{math|''l'' ≥ 2''n''}} such that {{math|''l'' {{!}} ''φ''(''a<sup>n</sup>'' − 1)}}.
*<math>\frac{\varphi(n)}{n}=\frac{\varphi(\operatorname{rad}(n))}{\operatorname{rad}(n)}</math>
::where {{math|rad(''n'')}} is the [[radical of an integer|radical of {{mvar|n}}]] (the product of all distinct primes dividing {{mvar|[[radical of an integer|n]]}}).
*<math>\sum_{d \mid n} \frac{\mu^2(d)}{\varphi(d)} = \frac{n}{\varphi(n)}</math>&nbsp;<ref>Dineva (in external refs), prop. 1</ref>
*<math>\sum_{1\le k\le n-1 \atop gcd(k,n)=1}\!\!k = \tfrac12 n\varphi(n) \quad \text{for }n>1</math>
*<math>\sum_{k=1}^n\varphi(k) = \tfrac12 \left(1+ \sum_{k=1}^n \mu(k)\left\lfloor\frac{n}{k}\right\rfloor^2\right)
=\frac3{\pi^2}n^2+O\left(n(\log n)^\frac23(\log\log n)^\frac43\right)</math>&nbsp;(<ref name=Wal1963>{{cite book | zbl=0146.06003 | last=Walfisz | first=Arnold | author-link=Arnold Walfisz | title=Weylsche Exponentialsummen in der neueren Zahlentheorie | language=de | series=Mathematische Forschungsberichte | volume=16 | location=Berlin | publisher=[[VEB Deutscher Verlag der Wissenschaften]] | year=1963 }}</ref> cited in<ref>{{citation | last = Lomadse | first = G. | title = The scientific work of Arnold Walfisz | journal = Acta Arithmetica | year = 1964 | volume = 10 | issue = 3 | pages = 227–237 | doi = 10.4064/aa-10-3-227-237 | url = http://matwbn.icm.edu.pl/ksiazki/aa/aa10/aa10111.pdf}}</ref>)
*<math>\sum_{k=1}^n\varphi(k) =\frac3{\pi^2}n^2+O\left(n(\log n)^\frac23(\log\log n)^\frac13\right) </math> [Liu (2016)]
*<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+O\left((\log n)^\frac23(\log\log n)^\frac43\right)</math>&nbsp;<ref name="Wal1963" />
*<math>\sum_{k=1}^n\frac{k}{\varphi(k)} = \frac{315\,\zeta(3)}{2\pi^4}n-\frac{\log n}2+O\left((\log n)^\frac23\right)</math>&nbsp;<ref name="Sita">{{cite journal|first=R. |last=Sitaramachandrarao |title=On an error term of Landau II |journal=Rocky Mountain J. Math. |volume=15 |date=1985 |issue=2 |pages=579–588|doi=10.1216/RMJ-1985-15-2-579 |doi-access=free }}</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)+O\left(\frac{(\log n)^\frac23}n\right)</math>&nbsp;<ref name="Sita" /><br>(where {{mvar|γ}} is the [[Euler–Mascheroni constant]]).

===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 {{math|[[Divisor function|''d''(''n'') {{=}} ''σ''<sub>0</sub>(''n'')]]}} is the number of divisors of {{mvar|n}}.

===Divisibility by any fixed positive integer===

The following property, which is unpublished as a specific result but has long been known,<ref>{{citation | last = Pollack | first = P. | title = Two problems on the distribution of Carmichael's lambda function | journal = Mathematika | year = 2023 | volume = 69 | issue =  4| pages = 1195–1220 | doi = 10.1112/mtk.12222
  | url =  | doi-access = free | arxiv = 2303.14043 }}</ref> has important consequences. For instance it rules out uniform distribution of the values of <math>\varphi(n)</math> in the arithmetic progressions modulo <math>q</math> for any integer <math>q>1</math>.

* For every fixed positive integer <math>q</math>, the relation <math>q|\varphi(n)</math> holds for almost all <math>n</math>, meaning for all but <math>o(x)</math> values of <math>n\le x</math> as <math>x\rightarrow\infty</math>.

This is an elementary consequence of the fact that the sum of the reciprocals of the primes congruent to 1 modulo <math>q</math> diverges, which itself is a corollary of the proof of [[Dirichlet's theorem on arithmetic progressions]].

==Generating functions==

The [[Dirichlet series]] for {{math|''φ''(''n'')}} may be written in terms of the [[Riemann zeta function]] as:<ref>{{harvnb|Hardy|Wright|1979|loc=thm. 288}}</ref>
:<math>\sum_{n=1}^\infty \frac{\varphi(n)}{n^s}=\frac{\zeta(s-1)}{\zeta(s)}</math>
where the left-hand side converges for <math>\Re (s)>2</math>.

The [[Lambert series]] generating function is<ref>{{harvnb|Hardy|Wright|1979|loc=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 {{math|{{abs|''q''}} < 1}}.

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

==Growth rate==

In the words of Hardy & Wright, the order of {{math|''φ''(''n'')}} is "always 'nearly {{mvar|n}}'."<ref>{{harvnb|Hardy|Wright|1979|loc=intro to § 18.4}}</ref>

First<ref>{{harvnb|Hardy|Wright|1979|loc=thm. 326}}</ref>

:<math>\lim\sup \frac{\varphi(n)}{n}= 1,</math>

but as ''n'' goes to infinity,<ref>{{harvnb|Hardy|Wright|1979|loc=thm. 327}}</ref> for all {{math|''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 {{math|''φ''(''n'')}} and the [[divisor function|divisor sum function]] {{math|''σ''(''n'')}}.

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 {{math|''n'' > 1}}, is proved.

We also have<ref name="hw328">{{harvnb|Hardy|Wright|1979|loc=thm. 328}}</ref>

:<math>\lim\inf\frac{\varphi(n)}{n}\log\log n = e^{-\gamma}.</math>

Here {{mvar|γ}} is [[Euler–Mascheroni constant|Euler's constant]], {{math|''γ'' {{=}} 0.577215665...}}, so {{math|''e<sup>γ</sup>'' {{=}} 1.7810724...}} and {{math|''e''<sup>−''γ''</sup> {{=}} 0.56145948...}}.

Proving this does not quite require the [[prime number theorem]].<ref>In fact Chebyshev's theorem ({{harvnb|Hardy|Wright|1979|loc=thm.7}}) and
Mertens' third theorem is all that is needed.</ref><ref>{{harvnb|Hardy|Wright|1979|loc=thm. 436}}</ref> Since {{math|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>Theorem 15 of {{cite journal|last1=Rosser |first1=J. Barkley |last2=Schoenfeld |first2=Lowell |title=Approximate formulas for some functions of prime numbers |journal=Illinois J. Math. |volume=6 |date=1962 |issue=1 |pages=64–94 |doi=10.1215/ijm/1255631807 |url=http://projecteuclid.org/euclid.ijm/1255631807|doi-access=free }}</ref><ref>Bach & Shallit, thm. 8.8.7</ref><ref name=Rib320>{{cite book|last=Ribenboim|title=The Book of Prime Number Records |edition=2nd |publisher=Springer-Verlag |location=New York |chapter=How are the Prime Numbers Distributed? §I.C The Distribution of Values of Euler's Function |pages=172–175 |doi= 10.1007/978-1-4684-0507-1_5 |date=1989 |isbn=978-1-4684-0509-5 }}</ref>

:<math>\varphi(n) > \frac {n} {e^\gamma\; \log \log n + \frac {3} {\log \log n}} \quad\text{for } n>2</math>

and

:<math>\varphi(n) < \frac {n} {e^{ \gamma}\log \log n} \quad\text{for infinitely many } n.</math>

The second inequality was shown by [[Jean-Louis Nicolas]]. 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/>{{rp|173}}

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}+O\left(n(\log n)^\frac23(\log\log n)^\frac43\right) \quad\text{as }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]]. 
By a combination of van der Corput's and Vinogradov's methods, H.-Q. Liu (On Euler's function.Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 4, 769–775)  
improved the error term to 
:<math>
O\left(n(\log n)^\frac23(\log\log n)^\frac13\right) 
</math>
(this is currently the best known estimate of this type). The [[Big O notation|"Big {{mvar|O}}"]] stands for a quantity that is bounded by a constant times the function of {{mvar|n}} inside the parentheses (which is small compared to {{math|''n''<sup>2</sup>}}).

This result can be used to prove<ref>{{harvnb|Hardy|Wright|1979|loc=thm. 332}}</ref> that [[Coprime integers#Probability of coprimality|the probability of two randomly chosen numbers being relatively prime]] is {{sfrac|6|{{pi}}<sup>2</sup>}}.

==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>\begin{align}
\lim\inf \frac{\varphi(n+1)}{\varphi(n)}&= 0 \quad\text{and} \\[5px]
\lim\sup \frac{\varphi(n+1)}{\varphi(n)}&= \infty.
\end{align}</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,\ldots\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,\ldots\right\}</math>

is dense in the interval (0,1).

==Totient number==
A '''totient number''' is a value of Euler's totient function: that is, an {{mvar|m}} for which there is at least one {{mvar|n}} for which {{math|''φ''(''n'') {{=}} ''m''}}. The ''valency'' or ''multiplicity'' of a totient number {{mvar|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. Every odd integer exceeding 1 is trivially a nontotient. There are also infinitely many even nontotients,<ref name=SC230>Sándor & Crstici (2004) p.230</ref> and indeed every positive integer has a multiple which is an even 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 | doi=10.1006/jnth.1993.1014| doi-access=free }}</ref>

The first few totient numbers are <math>1, 2, 4, 6, 8, 10, 12, 16, 18, 20</math>, see sequence  {{OEIS link|id=A002202}}.

The number of totient numbers up to a given limit {{mvar|x}} is

:<math>\frac{x}{\log x}e^{ \big(C+o(1)\big)(\log\log\log x)^2 } </math>

for a constant {{math|''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 | doi=10.1023/A:1009761909132 | issn=1382-4090 }} Reprinted in '' Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdos'', Developments in Mathematics, vol. 1, 1998, {{doi|10.1007/978-1-4757-4507-8_8}}, {{ISBN|978-1-4419-5058-1}}. Updated and corrected in {{arXiv|1104.3264}}, 2011.</ref>

If counted accordingly to multiplicity, the number of totient numbers up to a given limit {{mvar|x}} is

:<math>\Big\vert\{ n : \varphi(n) \le x \}\Big\vert = \frac{\zeta(2)\zeta(3)}{\zeta(6)} \cdot x + R(x)</math>

where the error term {{mvar|R}} is of order at most {{math|{{sfrac|''x''|(log ''x'')<sup>''k''</sup>}}}} for any positive {{mvar|k}}.<ref name=SMC22>Sándor et al (2006) p.22</ref>

It is known that the multiplicity of {{mvar|m}} exceeds {{math|''m''<sup>''δ''</sup>}} infinitely often for any {{math|''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 {{math|''k'' ≥ 2}} there is a totient number {{mvar|m}} of multiplicity {{mvar|k}}: that is, for which the equation {{math|''φ''(''n'') {{=}} ''m''}} has exactly {{mvar|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 {{mvar|m}} is known with multiplicity {{math|''k'' {{=}} 1}}. [[Carmichael's totient function conjecture]] is the statement that there is no such {{mvar|m}}.<ref name=SC228>Sándor & Crstici (2004) p.228</ref>

===Perfect totient numbers===

{{main|Perfect totient number}}
A perfect totient number is an integer that is equal to the sum of its iterated totients. That is, we apply the totient function to a number ''n'', apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals ''n'', then ''n'' is a perfect totient number.

==Applications==

===Cyclotomy===

{{main|Constructible polygon}}

In the last section of the [[Disquisitiones Arithmeticae|''Disquisitiones'']]<ref>Gauss, DA. The 7th § is arts. 336–366</ref><ref>Gauss proved if {{mvar|n}} satisfies certain conditions then the {{mvar|n}}-gon can be constructed. In 1837 [[Pierre Wantzel]] proved the converse, if the {{mvar|n}}-gon is constructible, then {{mvar|n}} must satisfy Gauss's conditions</ref> Gauss proves<ref>Gauss, DA, art 366</ref> that a regular {{mvar|n}}-gon can be constructed with straightedge and compass if {{math|''φ''(''n'')}} is a power of 2. If {{mvar|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 {{mvar|n}} is a first power and {{math|''n'' − 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 {{mvar|n}}-gon has a straightedge-and-compass construction if ''n'' is a product of distinct Fermat primes and any power of 2. The first few such {{mvar|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,... {{OEIS|A003401}}.

===Prime number theorem for arithmetic progressions===

{{main|Prime number theorem#Prime number theorem for arithmetic progressions}}

===The RSA cryptosystem===

{{main|RSA (algorithm)}}

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

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

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

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

==Unsolved problems==

===Lehmer's conjecture===

{{main|Lehmer's totient problem}}

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

In 1933 he proved that if any such {{mvar|n}} exists, it must be odd, square-free, and divisible by at least seven primes (i.e. {{math|''ω''(''n'') ≥ 7}}). In 1980 Cohen and Hagis proved that {{math|''n'' > 10<sup>20</sup>}} and that {{math|''ω''(''n'') ≥ 14}}.<ref>{{cite journal | zbl=0436.10002 | last1=Cohen | first1=Graeme L. | last2=Hagis | first2=Peter Jr. | title=On the number of prime factors of {{mvar|n}} if {{math|''φ''(''n'')}} divides {{math|''n'' − 1}} | journal=Nieuw Arch. Wiskd. |series=III Series | volume=28 | pages=177–185 | year=1980 | issn=0028-9825 }}</ref> Further, Hagis showed that if 3 divides {{mvar|n}} then {{math|''n'' > 10<sup>1937042</sup>}} and {{math|''ω''(''n'') ≥ 298848}}.<ref>{{cite journal | zbl=0668.10006 | last=Hagis | first=Peter Jr. | title=On the equation {{math|''M''·φ(''n'') {{=}} ''n'' − 1}} | journal=Nieuw Arch. Wiskd. |series=IV Series | 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 <math>n</math> with the property that for all other numbers <math>m</math>, <math>m\neq n</math>, <math>\varphi(m) \neq \varphi(n)</math>. See [[#Ford's theorem|Ford's theorem]] above.

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/>

===Riemann hypothesis===

The [[Riemann hypothesis]] is true [[if and only if]] the inequality
:<math>\frac{n}{\varphi (n)}<e^\gamma \log\log n+\frac{e^\gamma (4+\gamma-\log 4\pi)}{\sqrt{\log n}}</math>
is true for all <math>n\geq p_{120569}\#</math> where <math>\gamma</math> is [[Euler's constant]] and <math>p_{120569}\#</math> is the [[Primorial|product of the first]] {{math|120569}} primes.<ref>{{Cite book |last1=Broughan |first1=Kevin |title=Equivalents of the Riemann Hypothesis, Volume One: Arithmetic Equivalents |publisher=Cambridge University Press |year=2017 |edition=First |isbn=978-1-107-19704-6}} Corollary 5.35</ref>

== See also ==
*[[Carmichael function|Carmichael function (λ)]]
*[[Dedekind psi function|Dedekind psi function (𝜓)]]
*[[Divisor function|Divisor function (σ)]]
*[[Duffin–Schaeffer conjecture]]
*[[Fermat's little theorem#Generalizations|Generalizations of Fermat's little theorem]]
*[[Highly composite number]]
*[[Multiplicative group of integers modulo n|Multiplicative group of integers modulo {{mvar|n}}]]
*[[Ramanujan sum]]
*[[Totient summatory function|Totient summatory function (𝛷)]]

== Notes ==
{{Reflist|30em}}

==References==

{{refbegin|colwidth=30em}}

The ''[[Disquisitiones Arithmeticae]]'' has been translated from Latin into English and German. The German edition includes all of Gauss's 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''.

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* Dickson, Leonard Eugene, "History Of The Theory Of Numbers", vol 1, chapter 5 "Euler's Function, Generalizations; Farey Series", Chelsea Publishing 1952
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*{{citation
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* {{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
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* {{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}}
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{{refend}}

==External links==<!-- This section is linked from [[Euler's totient function]] -->
* {{springer|title=Totient function|id=p/t110040}}
*[http://mathcenter.oxford.emory.edu/site/math125/chineseRemainderTheorem/ Euler's Phi Function and the Chinese Remainder Theorem — proof that {{math|''φ''(''n'')}} is multiplicative] {{Webarchive|url=https://web.archive.org/web/20210228071226/http://mathcenter.oxford.emory.edu/site/math125/chineseRemainderTheorem/ |date=2021-02-28 }}
*[http://www.javascripter.net/math/calculators/eulertotientfunction.htm Euler's totient function calculator in JavaScript — up to 20 digits]
*Dineva, Rosica, [http://www.mtholyoke.edu/~robinson/reu/reu05/rdineva1.pdf The Euler Totient, the Möbius, and the Divisor Functions] {{Webarchive|url=https://web.archive.org/web/20210116061553/https://www.mtholyoke.edu/~robinson/reu/reu05/rdineva1.pdf |date=2021-01-16 }}
*Plytage, Loomis, Polhill [http://facstaff.bloomu.edu/jpolhill/cmj034-042.pdf Summing Up The Euler Phi Function]

{{Totient}}

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