View source for Euler's totient function

{{Redirect|Φ(n)||phi}}

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

For example, the totatives of {{math|1= ''n'' = 9}} 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, because {{math|1= gcd(9, 3) = gcd(9, 6) = 3}} and {{math|1= gcd(9, 9) = 9}}. Therefore, {{math|1= ''φ''(9) = 6}}. As another example, {{math|1= ''φ''(1) = 1}} since for {{math|1= ''n'' = 1}} the only integer in the range from 1 to {{mvar|n}} is 1 itself, and {{math|1= gcd(1, 1) = 1}}.

Euler's totient function is a [[multiplicative function]], meaning that if two numbers {{mvar|m}} and {{mvar|n}} are relatively prime, then {{math|1= ''φ''(''mn'') = ''φ''(''m'')''φ''(''n'')}}.<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'''}}).<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 {{mvar|n}} as the number of integers that are smaller than {{mvar|N}} and relatively prime to {{mvar|N}} (… 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 {{mvar|π}} to denote it: he wrote {{math|''πD''}} for "the multitude of numbers less than {{mvar|D}}, 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|1=''D'' = 1}} 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|''φ''(''A'')}} comes from [[Gauss]]'s 1801 treatise ''[[Disquisitiones Arithmeticae]]'',<ref>Gauss, ''Disquisitiones Arithmeticae'' article&nbsp;38</ref> although Gauss didn't use parentheses around the argument and wrote {{math|''φA''}}. 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#v=onepage&q&f=false 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'''. [[Jordan's totient function|Jordan's totient]] is a generalization of Euler's.

The '''cototient''' of {{mvar|n}} is defined as {{math|''n'' − ''φ''(''n'')}}. It counts the number of positive integers less than or equal to {{mvar|n}} that have at least one [[prime number|prime factor]] in common with {{mvar|n}}.

== Computing Euler's totient function ==

There are several formulas for computing {{math|''φ''(''n'')}}.

===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}}. (The notation is described in the article [[Arithmetical function#Notation|Arithmetical function]].)

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

==== The function is multiplicative ====

This means that if {{math|1=gcd(''m'', ''n'') = 1}}, then {{math|1=''φ''(''mn'') = ''φ''(''m'') ''φ''(''n'')}}. (''Outline of proof'': let {{mvar|A}}, {{mvar|B}}, {{mvar|C}} be the sets of nonnegative integers, which are, respectively, [[coprime]] to and less than {{mvar|m}}, {{mvar|n}}, and {{mvar|mn}}; then there is a [[bijection]] between {{math|''A'' × ''B''}} and {{mvar|C}}, by the [[Chinese remainder theorem]].)

==== Value for a prime power argument ====

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

:<math>\varphi \left(p^k\right) = p^{k-1}(p-1) = p^k \left( 1 - \frac{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 for {{math|gcd(''p''<sup>''k''</sup>, ''m'')}} to not equal 1 is for {{mvar|m}} to be a multiple of {{mvar|p}}. The multiples of {{mvar|p}} that are less than or equal to {{math|''p''<sup>''k''</sup>}} are {{math|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>}}{{math|}} of them. 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 for {{mvar|n}},

:<math>n = {p_1}^{k_1} \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{align} 
\varphi(n)&= \varphi\left(p_1^{k_1}\right) \varphi\left(p_2^{k_2}\right) \cdots\varphi\left(p_r^{k_r}\right)\\
&= 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-\tfrac12\right)\left(1-\tfrac13\right)=36\cdot\tfrac12\cdot\tfrac23=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.<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''}|}}. 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 {2\pi\frac{k}{n}}.</math>

Unlike the other two formulae (the Euler product and the divisor sum) 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 way 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''⟩}}, 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 all subgroups of {{math|''C''<sub>''d''</sub> ≤ ''C''<sub>''n''</sub>}} are generated by some element of {{math|''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 {{mvar|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 {{math|''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. Which fractions have 20 as denominator? The ones whose numerators are relatively prime to 20 ({{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) {{=}} 4}} fractions with denominator 10 ({{sfrac|1|10}}, {{sfrac|3|10}}, {{sfrac|7|10}}, {{sfrac|9|10}}), {{math|''φ''(5) {{=}} 4}} fractions with denominator 5 ({{sfrac|1|5}}, {{sfrac|2|5}}, {{sfrac|3|5}}, {{sfrac|4|5}}), and so on.

In detail, we are considering the fractions of the form {{mvar|{{sfrac|k|n}}}} where {{mvar|k}} is an integer from 1 to {{mvar|n}} inclusive. Upon reducing these to lowest terms, each fraction will have as its denominator some divisor of {{mvar|n}}. We can group the fractions together by denominator, and we must show that for a given divisor {{mvar|d}} of {{mvar|n}}, the number of such fractions with denominator {{mvar|d}} is {{math|''φ''(''d'')}}.

Note that to reduce {{mvar|{{sfrac|k|n}}}} to lowest terms, we divide the numerator and denominator by {{math|gcd(''k'', ''n'')}}. The reduced fractions with denominator {{mvar|d}} are therefore precisely the ones originally of the form {{mvar|{{sfrac|k|n}}}} in which {{math|gcd(''k'', ''n'') {{=}} ''{{sfrac|n|d}}''}}. The question therefore becomes: how many {{mvar|k}} are there less than or equal to {{mvar|n}} which verify {{math|gcd(''k'', ''n'') {{=}} ''{{sfrac|n|d}}''}}? Any such {{mvar|k}} must clearly be a multiple of {{mvar|{{sfrac|n|d}}}}, but it must also be coprime to {{mvar|d}} (if it had any common divisor {{mvar|s}} with {{mvar|d}}, then {{mvar|{{sfrac|sn|d}}}} would be a larger common divisor of {{mvar|n}} and {{mvar|k}}). Conversely, any multiple {{mvar|k}} of {{mvar|{{sfrac|n|d}}}} which is coprime to {{mvar|d}} will satisfy {{math|gcd(''k'', ''n'') {{=}} ''{{sfrac|n|d}}''}}. We can generate {{math|''φ''(''d'')}} such numbers by taking the numbers less than {{mvar|d}} coprime to {{mvar|d}} and multiplying each one by {{mvar|{{sfrac|n|d}}}} (these products will of course each be smaller than {{mvar|n}}, as required). This in fact generates all such numbers, as if {{mvar|k}} is a multiple of {{mvar|{{sfrac|n|d}}}} coprime to {{mvar|d}} (and less than {{mvar|n}}), then {{math|{{sfrac|''k''|{{frac|''n''|''d''}}}}}} will still be coprime to {{mvar|d}}, and must also be smaller than {{mvar|d}}, else {{mvar|k}} would be larger than {{mvar|n}}. Thus there are precisely {{math|''φ''(''d'')}} values of {{mvar|k}} less than or equal to {{mvar|n}} such that {{math|gcd(''k'', ''n'') {{=}} ''{{sfrac|n|d}}''}}, which was to be demonstrated.

[[Möbius inversion]] 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]].

This formula may also be derived from the product formula by multiplying out

:<math> \prod_{p\mid n} \left(1 - \frac{1}{p}\right) </math>

to get

:<math> \sum_{d\mid n} \frac{\mu (d)}{d}. </math>

==Some values of the function==

The first 143 values {{OEIS|A000010}} are shown in the table and graph below:<ref>The cell for {{math|''n'' {{=}} 0}} in the upper-left corner of the table is empty, as the function {{math|''φ''(''n'')}} is commonly defined only for positive integers, so it is not defined for {{math|''n'' {{=}} 0}}.</ref>

[[File:EulerPhi100.svg|thumb|Graph of the first 100 values]]
:{| class="wikitable" style="text-align: right"
|+{{math|''φ''(''n'')}} for {{math|1 ≤ ''n'' ≤ 143}}
! +
! 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11
|- 
! 0 
| {{n/a}} || 1 || 1 || 2 || 2 || 4 || 2 || 6 || 4 || 6 || 4 || 10
|-
! 12
| 4 || 12 || 6 || 8 || 8 || 16 || 6 || 18 || 8 || 12 || 10 || 22
|- 
! 24
| 8 || 20 || 12 || 18 || 12 || 28 || 8 || 30 || 16 || 20 || 16 || 24
|-
! 36
| 12 || 36 || 18 || 24 || 16 || 40 || 12 || 42 || 20 || 24 || 22 || 46
|-
! 48
| 16 || 42 || 20 || 32 || 24 || 52 || 18 || 40 || 24 || 36 || 28 || 58
|-
! 60
| 16 || 60 || 30 || 36 || 32 || 48 || 20 || 66 || 32 || 44 || 24 || 70
|-
! 72
| 24 || 72 || 36 || 40 || 36 || 60 || 24 || 78 || 32 || 54 || 40 || 82
|-
! 84
| 24 || 64 || 42 || 56 || 40 || 88 || 24 || 72 || 44 || 60 || 46 || 72
|-
! 96
| 32 || 96 || 42 || 60 || 40 || 100 || 32 || 102 || 48 || 48 || 52 || 106
|-
! 108
| 36 || 108 || 40 || 72 || 48 || 112 || 36 || 88 || 56 || 72 || 58 || 96
|-
! 120
| 32 || 110 || 60 || 80 || 60 || 100 || 36 || 126 || 64 || 84 || 48 || 130
|-
! 132
| 40 || 108 || 66 || 72 || 64 || 136 || 44 || 138 || 48 || 92 || 70 || 120
|}

The top line in the graph, {{math|''y'' {{=}} ''n'' − 1}}, is a true [[upper bound]]. It is attained whenever {{mvar|n}} is prime. 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. More precisely, the lower limit of the graph is proportional to {{math|{{sfrac|''n''|log log ''n''}}}} rather than being linear.<ref name="hw328"/>
{{clear}}

==Euler's theorem==

{{main article|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> n \mid \varphi(a^n-1) \quad \text{for } a,n > 1</math>
*<math>\varphi(mn) = \varphi(m)\varphi(n)\cdot\frac{d}{\varphi(d)} \quad\text{where }d = \operatorname{gcd}(m,n)</math>
: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>
:*<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>\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}}. 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}}]].
*<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 \atop (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 | 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> cited in<ref>{{citation | last = Lomadse | first = G. | title = The scientific work of Arnold Walfisz | journal = Acta Arithmetica | volume = 10 | issue = 3 | pages = 227–237 | url = http://matwbn.icm.edu.pl/ksiazki/aa/aa10/aa10111.pdf}}</ref>)

*<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 |pages=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)+O\left(\frac{(\log n)^\frac23}n\right)</math>&nbsp;<ref name=Sita />

:(where {{mvar|γ}} is the [[Euler–Mascheroni constant]]).

*<math>\sum_\stackrel{1\le k\le n}{\operatorname{gcd}(k,m)=1} \!\!\!\! 1 = n \frac {\varphi(m)}{m} + O \left ( 2^{\omega(m)} \right )</math>
:where {{math|''m'' > 1}} is a positive integer and {{math|''ω''(''m'')}} is the number of distinct prime factors of {{mvar|m}}.<ref>Bordellès in the [[#External links|external links]]</ref>

===Menon's identity===

{{Main article|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|{{math|''d''(''n'') {{=}} ''σ''<sub>0</sub>(''n'')}}]] is the number of divisors of {{mvar|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|''μ''(''n'')}}. In this section {{math|''φ''(''n'')}} is the totient function, and {{math|''ϕ'' {{=}} {{sfrac|1 + {{sqrt|5}}|2}} {{=}} 1.618…}} 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 [[e (mathematical 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 two formulae
:<math>\begin{align}
\sum_{k=1}^\infty\frac{\varphi(k)}{k}\left(-\log\left(1-x^k\right)\right)&=\frac{x}{1-x} \\
\text{and}\; \sum_{k=1}^\infty\frac{\mu(k)}{k}\left(-\log\left(1-x^k\right)\right)&=x, \qquad \quad \text{for } 0<x<1.
\end{align}</math>

==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>

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 |url=http://projecteuclid.org/euclid.ijm/1255631807}}</ref><ref>Bach & Shallit, thm. 8.8.7</ref><ref name=Rib320>{{cite book|last=Ribenboim|title=The Book of Prime Number Records|at=Section 4.I.C}}{{full citation needed|date=July 2017}}</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/>

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]] (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 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,\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 {{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}}</ref>

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 | issn=1382-4090 | doi=10.1007/978-1-4757-4507-8_8| arxiv=1104.3264 }}</ref>

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

:<math>\Big\vert\{ n : \phi(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 article|Perfect totient number}}

==Applications==

===Cyclotomy===

{{main article|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}}.

===The RSA cryptosystem===

{{main article|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 factored or if {{math|''φ''(''n'')}} could be computed without factoring {{mvar|n}}.

==Unsolved problems==

===Lehmer's conjecture===

{{main article|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'') {{!}} ''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., III. Ser. | 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., 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 article|Carmichael's totient function conjecture}}

This states that there is no number {{mvar|n}} with the property that for all other numbers {{mvar|m}}, {{math|''m'' ≠ ''n''}}, {{math|''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|Multiplicative group of integers modulo {{mvar|n}}]]
*[[Ramanujan sum]]
*[[Totient summatory function]]

== 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' 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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{{refend}}

==External links==<!-- This section is linked from [[Euler's totient function]] -->
* {{springer|title=Totient function|id=p/t110040}}
*[http://www.oxfordmathcenter.com/drupal7/node/172 Euler's Phi Function and the Chinese Remainder Theorem — proof that {{math|''φ''(''n'')}} is multiplicative]
*[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]
*Plytage, Loomis, Polhill [http://facstaff.bloomu.edu/jpolhill/cmj034-042.pdf Summing Up The Euler Phi Function]

{{Totient}}

[[Category:Modular arithmetic]]
[[Category:Multiplicative functions]]
[[Category:Articles containing proofs]]
[[Category:Algebra]]
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[[Category:Leonhard Euler]]