:Euler's totient function

Leonhard Euler introduced the function in 1763.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).Sandifer, p. 203Graham et al. p. 133 note 111 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".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). This definition varies from the current definition for the totient function at {{math|1=D = 1}} but is otherwise the same. The now-standard notationBoth {{math|φ(n)}} and {{math|ϕ(n)}} are seen in the literature. These are two forms of the lower-case Greek letter phi. {{math|φ(A)}} comes from Gauss's 1801 treatise Disquisitiones Arithmeticae,Gauss, Disquisitiones Arithmeticae article 38{{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}} although Gauss did not 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, J. J. Sylvester coined the term totient for this function,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].{{cite OED2|totient}} so it is also referred to as Euler's totient function, the Euler totient, or Euler's totient.{{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}} 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 factor in common with {{mvar|n}}.

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

It states

:$\backslash varphi(n)\; =n\; \backslash prod\_\{p\backslash mid\; n\}\; \backslash left(1-\backslash frac\{1\}\{p\}\backslash right),$

where the product is over the distinct prime numbers dividing {{mvar|n}}. (For notation, see Arithmetical function.)

An equivalent formulation is

$$\backslash varphi(n)\; =\; p\_1^\{k\_1-1\}(p\_1\{-\}1)\backslash ,p\_2^\{k\_2-1\}(p\_2\{-\}1)\backslash cdots\; p\_r^\{k\_r-1\}(p\_r\{-\}1),$$ where $n\; =\; p\_1^\{k\_1\}\; p\_2^\{k\_2\}\; \backslash cdots\; p\_r^\{k\_r\}$ is the prime factorization of $n$ (that is, $p\_1,\; p\_2,\backslash ldots,p\_r$ are distinct prime numbers).

The proof of these formulae depends on two important facts.

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.

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

:$\backslash varphi\; \backslash left(p^k\backslash right)\; =\; p^k-p^\{k-1\}\; =\; p^\{k-1\}(p-1)\; =\; p^k\; \backslash left(\; 1\; -\; \backslash tfrac\{1\}\{p\}\; \backslash right).$

Proof: Since {{mvar|p}} is a prime number, the only possible values of {{math|gcd(p^k, m)}} are {{math|1, p, p², ..., p^k}}, and the only way to have {{math|gcd(p^k, m) > 1}} is if {{mvar|m}} is a multiple of {{mvar|p}}, that is, {{math|1=m ∈ {{mset|1=p, 2p, 3p, ..., p^{k − 1}p = p^k}}}}, and there are {{math|p^{k − 1}}} such multiples not greater than {{math|p^k}}. Therefore, the other {{math|p^k − p^{k − 1}}} numbers are all relatively prime to {{math|p^k}}.

The fundamental theorem of arithmetic states that if {{math|n > 1}} there is a unique expression $n\; =\; p\_1^\{k\_1\}\; p\_2^\{k\_2\}\; \backslash cdots\; p\_r^\{k\_r\},$ where {{math|p₁ < p₂ < ... < p_r}} are prime numbers and each {{math|k_i ≥ 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^k)}} gives

:$\backslash begin\{array\}\; \{rcl\}$

\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}

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 $\backslash \{1,2,\backslash ldots,n\backslash \}$, excluding the sets of integers divisible by the prime divisors.

:$\backslash varphi(20)=\backslash varphi(2^2\; 5)=20\backslash ,(1-\backslash tfrac12)\backslash ,(1-\backslash tfrac15)$

=20\cdot\tfrac12\cdot\tfrac45=8.

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:$$\backslash varphi(20)\; =\; \backslash varphi(2^2\; 5^1)=\; 2^\{2-1\}(2\{-\}1)\backslash ,5^\{1-1\}(5\{-\}1)\; =\; 2\backslash cdot\; 1\backslash cdot\; 1\backslash cdot\; 4\; =\; 8.$$

The totient is the discrete Fourier transform of the gcd, evaluated at 1.{{harvtxt|Schramm|2008}} Let

:$\backslash mathcal\{F\}\; \backslash \{\; \backslash mathbf\{x\}\; \backslash \}[m]\; =\; \backslash sum\backslash limits\_\{k=1\}^n\; x\_k\; \backslash cdot\; e^\{\{-2\backslash pi\; i\}\backslash frac\{mk\}\{n\}\}$

where {{math|x_k {{=}} gcd(k,n)}} for {{math|k ∈ {1, ..., n}}}. Then

:$\backslash varphi\; (n)\; =\; \backslash mathcal\{F\}\; \backslash \{\; \backslash mathbf\{x\}\; \backslash \}[1]\; =\; \backslash sum\backslash limits\_\{k=1\}^n\; \backslash gcd(k,n)\; e^\{-2\backslash pi\; i\backslash frac\{k\}\{n\}\}.$

The real part of this formula is

:$\backslash varphi\; (n)=\backslash sum\backslash limits\_\{k=1\}^n\; \backslash gcd(k,n)\; \backslash cos\; \{\backslash tfrac\{2\backslash pi\; k\}\{n\}\}$

.

For example, using $\backslash cos\backslash tfrac\{\backslash pi\}5\; =\; \backslash tfrac\{\backslash sqrt\; 5+1\}4$

and $\backslash cos\backslash tfrac\{2\backslash pi\}5\; =\; \backslash tfrac\{\backslash sqrt\; 5-1\}4$

:$$\backslash begin\{array\}\{rcl\}$$

\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}

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.

The property established by Gauss,Gauss, DA, art 39 that

:$\backslash sum\_\{d\backslash mid\; n\}\backslash varphi(d)=n,$

where the sum is over all positive divisors {{mvar|d}} of {{mvar|n}}, can be proven in several ways. (See 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_d}} ; specifically, if {{math|C_d {{=}} ⟨g⟩}} with {{math|1=g^d = 1}}, then {{math|g^k}} is a generator for every {{mvar|k}} coprime to {{mvar|d}}. Since every element of {{math|C_n}} generates a cyclic subgroup, and each subgroup {{math|C_d ⊆ C_n}} is generated by precisely {{math|φ(d)}} elements of {{math|C_n}}, the formula follows.Gauss, DA art. 39, arts. 52-54 Equivalently, the formula can be derived by the same argument applied to the Root of unity#Group of nth roots of unity and the primitive root of unity.

The formula can also be derived from elementary arithmetic.Graham et al. pp. 134-135 For example, let {{math|n {{=}} 20}} and consider the positive fractions up to 1 with denominator 20:

:$$

\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}.

Put them into lowest terms:

:$$

\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}

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

:$\backslash varphi(n)\; =\; \backslash sum\_\{d\backslash mid\; n\}\; \backslash mu\backslash left(\; d\; \backslash right)\; \backslash cdot\; \backslash frac\{n\}\{d\}\; =\; n\backslash sum\_\{d\backslash mid\; n\}\; \backslash frac\{\backslash mu\; (d)\}\{d\},$

where {{mvar|μ}} is the Möbius function, the multiplicative function defined by $\backslash mu(p)\; =\; -1$ and $\backslash mu(p^k)\; =\; 0$ for each prime {{math|1=p}} and {{math|1=k ≥ 2}}. This formula may also be derived from the product formula by multiplying out $\backslash prod\_\{p\backslash mid\; n\}\; (1\; -\; \backslash frac\{1\}\{p\})$ to get $\backslash sum\_\{d\; \backslash mid\; n\}\; \backslash frac\{\backslash mu\; (d)\}\{d\}.$

An example:$$$$

\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}

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

File:EulerPhi100.svg

:

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 $\backslash varphi(n)\; \backslash ge\; \backslash sqrt\{n/2\}$, which is rather loose: in fact, the lower limit of the graph is proportional to {{math|{{sfrac|n|log log n}}}}.

{{clear}}

{{main article|Euler's theorem}}

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

:$a^\{\backslash varphi(n)\}\; \backslash equiv\; 1\backslash mod\; n.$

The special case where {{mvar|n}} is prime is known as Fermat's little theorem.

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

The RSA cryptosystem is based on this theorem: it implies that the inverse of the function {{math|a ↦ a^e mod n}}, where {{mvar|e}} is the (public) encryption exponent, is the function {{math|b ↦ b^d 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 difficulty to factor large numbers we have the guarantee that no one else knows the factorization.

• $a\backslash mid\; b\; \backslash implies\; \backslash varphi(a)\backslash mid\backslash varphi(b)$
• $m\; \backslash mid\; \backslash varphi(a^m-1)$
• $\backslash varphi(mn)\; =\; \backslash varphi(m)\backslash varphi(n)\backslash cdot\backslash frac\{d\}\{\backslash varphi(d)\}\; \backslash quad\backslash text\{where\; \}d\; =\; \backslash operatorname\{gcd\}(m,n)$

  In particular:

  • $\backslash varphi(2m)\; =\; \backslash begin\{cases\}$

  2\varphi(m) &\text{ if } m \text{ is even} \\

  \varphi(m) &\text{ if } m \text{ is odd}

  \end{cases}

  • $\backslash varphi\backslash left(n^m\backslash right)\; =\; n^\{m-1\}\backslash varphi(n)$

• $\backslash varphi(\backslash operatorname\{lcm\}(m,n))\backslash cdot\backslash varphi(\backslash operatorname\{gcd\}(m,n))\; =\; \backslash varphi(m)\backslash cdot\backslash varphi(n)$

  Compare this to the formula $\backslash operatorname\{lcm\}(m,n)\backslash cdot\; \backslash operatorname\{gcd\}(m,n)\; =\; m\; \backslash cdot\; n$ (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^r {{!}} φ(n)}}

• For any {{math|a > 1}} and {{math|n > 6}} such that {{math|4 ∤ n}} there exists an {{math|l ≥ 2n}} such that {{math|l {{!}} φ(aⁿ − 1)}}.
• $\backslash frac\{\backslash varphi(n)\}\{n\}=\backslash frac\{\backslash varphi(\backslash operatorname\{rad\}(n))\}\{\backslash operatorname\{rad\}(n)\}$

  where {{math|rad(n)}} is the radical of an integer (the product of all distinct primes dividing {{mvar|n}}).

• $\backslash sum\_\{d\; \backslash mid\; n\}\; \backslash frac\{\backslash mu^2(d)\}\{\backslash varphi(d)\}\; =\; \backslash frac\{n\}\{\backslash varphi(n)\}$ Dineva (in external refs), prop. 1
• $\backslash sum\_\{1\backslash le\; k\backslash le\; n-1\; \backslash atop\; gcd(k,n)=1\}\backslash !\backslash !k\; =\; \backslash tfrac12\; n\backslash varphi(n)\; \backslash quad\; \backslash text\{for\; \}n>1$
• $\backslash sum\_\{k=1\}^n\backslash varphi(k)\; =\; \backslash tfrac12\; \backslash left(1+\; \backslash sum\_\{k=1\}^n\; \backslash mu(k)\backslash left\backslash lfloor\backslash frac\{n\}\{k\}\backslash right\backslash rfloor^2\backslash right)$

  =\frac3{\pi^2}n^2+O\left(n(\log n)^\frac23(\log\log n)^\frac43\right) ({{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 }} cited in{{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}})

• $\backslash sum\_\{k=1\}^n\backslash varphi(k)\; =\backslash frac3\{\backslash pi^2\}n^2+O\backslash left(n(\backslash log\; n)^\backslash frac23(\backslash log\backslash log\; n)^\backslash frac13\backslash right)$ [Liu (2016)]
• $\backslash sum\_\{k=1\}^n\backslash frac\{\backslash varphi(k)\}\{k\}\; =\; \backslash sum\_\{k=1\}^n\backslash frac\{\backslash mu(k)\}\{k\}\backslash left\backslash lfloor\backslash frac\{n\}\{k\}\backslash right\backslash rfloor=\backslash frac6\{\backslash pi^2\}n+O\backslash left((\backslash log\; n)^\backslash frac23(\backslash log\backslash log\; n)^\backslash frac43\backslash right)$ 
• $\backslash sum\_\{k=1\}^n\backslash frac\{k\}\{\backslash varphi(k)\}\; =\; \backslash frac\{315\backslash ,\backslash zeta(3)\}\{2\backslash pi^4\}n-\backslash frac\{\backslash log\; n\}2+O\backslash left((\backslash log\; n)^\backslash frac23\backslash right)$ {{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 }}
• $\backslash sum\_\{k=1\}^n\backslash frac\{1\}\{\backslash varphi(k)\}\; =\; \backslash frac\{315\backslash ,\backslash zeta(3)\}\{2\backslash pi^4\}\backslash left(\backslash log\; n+\backslash gamma-\backslash sum\_\{p\backslash text\{\; prime\}\}\backslash frac\{\backslash log\; p\}\{p^2-p+1\}\backslash right)+O\backslash left(\backslash frac\{(\backslash log\; n)^\backslash frac23\}n\backslash right)$ 

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

{{Main article|Arithmetic_function#Menon.27s_identity|l1=Menon's identity}}

In 1965 P. Kesava Menon proved

:$\backslash sum\_\{\backslash stackrel\{1\backslash le\; k\backslash le\; n\}\{\; \backslash gcd(k,n)=1\}\}\; \backslash !\backslash !\backslash !\backslash !\; \backslash gcd(k-1,n)=\backslash varphi(n)d(n),$

where {{math|d(n) {{=}} σ₀(n)}} is the number of divisors of {{mvar|n}}.

The following property, which is part of the « folklore » (i.e., apparently unpublished as a specific result: {{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 }} see the introduction of this article in which it is stated as having « long been known ») has important consequences. For instance it rules out uniform distribution of the values of $\backslash varphi(n)$ in the arithmetic progressions modulo $q$ for any integer $q>1$.

• For every fixed positive integer $q$, the relation $q|\backslash varphi(n)$ holds for almost all $n$, meaning for all but $o(x)$ values of $n\backslash le\; x$ as $x\backslash rightarrow\backslash infty$.

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

The Dirichlet series for {{math|φ(n)}} may be written in terms of the Riemann zeta function as:{{harvnb|Hardy|Wright|1979|loc=thm. 288}}

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash varphi(n)\}\{n^s\}=\backslash frac\{\backslash zeta(s-1)\}\{\backslash zeta(s)\}$

where the left-hand side converges for $\backslash Re\; (s)>2$.

The Lambert series generating function is{{harvnb|Hardy|Wright|1979|loc=thm. 309}}

:$\backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{\backslash varphi(n)\; q^n\}\{1-q^n\}=\; \backslash frac\{q\}\{(1-q)^2\}$

which converges for {{math|{{abs|q}} < 1}}.

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

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

First{{harvnb|Hardy|Wright|1979|loc=thm. 326}}

:$\backslash lim\backslash sup\; \backslash frac\{\backslash varphi(n)\}\{n\}=\; 1,$

but as n goes to infinity,{{harvnb|Hardy|Wright|1979|loc=thm. 327}} for all {{math|δ > 0}}

:$\backslash frac\{\backslash varphi(n)\}\{n^\{1-\backslash delta\}\}\backslash rightarrow\backslash infty.$

These two formulae can be proved by using little more than the formulae for {{math|φ(n)}} and the divisor sum function {{math|σ(n)}}.

In fact, during the proof of the second formula, the inequality

:

true for {{math|n > 1}}, is proved.

We also have{{harvnb|Hardy|Wright|1979|loc=thm. 328}}

:$\backslash lim\backslash inf\backslash frac\{\backslash varphi(n)\}\{n\}\backslash log\backslash log\; n\; =\; e^\{-\backslash gamma\}.$

Here {{mvar|γ}} is Euler's constant, {{math|γ {{=}} 0.577215665...}}, so {{math|e^γ {{=}} 1.7810724...}} and {{math|e^−γ {{=}} 0.56145948...}}.

Proving this does not quite require the prime number theorem.In fact Chebyshev's theorem ({{harvnb|Hardy|Wright|1979|loc=thm.7}}) and

Mertens' third theorem is all that is needed.{{harvnb|Hardy|Wright|1979|loc=thm. 436}} Since {{math|log log n}} goes to infinity, this formula shows that

:$\backslash lim\backslash inf\backslash frac\{\backslash varphi(n)\}\{n\}=\; 0.$

In fact, more is true.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 }}Bach & Shallit, thm. 8.8.7{{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 }}

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

and

:

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."{{rp|173}}

For the average order, we haveSándor, Mitrinović & Crstici (2006) pp.24–25

:$\backslash varphi(1)+\backslash varphi(2)+\backslash cdots+\backslash varphi(n)\; =\; \backslash frac\{3n^2\}\{\backslash pi^2\}+O\backslash left(n(\backslash log\; n)^\backslash frac23(\backslash log\backslash log\; n)^\backslash frac43\backslash right)\; \backslash quad\backslash text\{as\; \}n\backslash rightarrow\backslash infty,$

due to Arnold Walfisz, its proof exploiting estimates on exponential sums due to 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

:$$

O\left(n(\log n)^\frac23(\log\log n)^\frac13\right)

(this is currently the best known estimate of this type). The Big O notation 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|n²}}).

This result can be used to prove{{harvnb|Hardy|Wright|1979|loc=thm. 332}} that the probability of two randomly chosen numbers being relatively prime is {{sfrac|6|{{pi}}²}}.

In 1950 Somayajulu provedRibenboim, p.38Sándor, Mitrinović & Crstici (2006) p.16

:$\backslash begin\{align\}$

\lim\inf \frac{\varphi(n+1)}{\varphi(n)}&= 0 \quad\text{and} \\[5px]

\lim\sup \frac{\varphi(n+1)}{\varphi(n)}&= \infty.

\end{align}

In 1954 Schinzel and Sierpiński strengthened this, proving that the set

:$\backslash left\backslash \{\backslash frac\{\backslash varphi(n+1)\}\{\backslash varphi(n)\},\backslash ;\backslash ;n\; =\; 1,2,\backslash ldots\backslash right\backslash \}$

is dense in the positive real numbers. They also proved that the set

:$\backslash left\backslash \{\backslash frac\{\backslash varphi(n)\}\{n\},\backslash ;\backslash ;n\; =\; 1,2,\backslash ldots\backslash right\backslash \}$

is dense in the interval (0,1).

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.Guy (2004) p.144 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,Sándor & Crstici (2004) p.230 and indeed every positive integer has a multiple which is an even nontotient.{{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 }}

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

:$\backslash frac\{x\}\{\backslash log\; x\}e^\{\; \backslash big(C+o(1)\backslash big)(\backslash log\backslash log\backslash log\; x)^2\; \}$

for a constant {{math|C {{=}} 0.8178146...}}.{{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.

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

:$\backslash Big\backslash vert\backslash \{\; n\; :\; \backslash varphi(n)\; \backslash le\; x\; \backslash \}\backslash Big\backslash vert\; =\; \backslash frac\{\backslash zeta(2)\backslash zeta(3)\}\{\backslash zeta(6)\}\; \backslash cdot\; x\; +\; R(x)$

where the error term {{mvar|R}} is of order at most {{math|{{sfrac|x|(log x)^k}}}} for any positive {{mvar|k}}.Sándor et al (2006) p.22

It is known that the multiplicity of {{mvar|m}} exceeds {{math|m^δ}} infinitely often for any {{math|δ < 0.55655}}.Sándor et al (2006) p.21Guy (2004) p.145

{{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,Sándor & Crstici (2004) p.229 and it had been obtained as a consequence of Schinzel's hypothesis H. Indeed, each multiplicity that occurs, does so infinitely often.

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}}.Sándor & Crstici (2004) p.228

{{main article|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.

{{main article|Constructible polygon}}

In the last section of the DisquisitionesGauss, DA. The 7th § is arts. 336–366Gauss 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 Gauss provesGauss, DA, art 366 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 primes, 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}} areGauss, DA, art. 366. This list is the last sentence in the Disquisitiones

:2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40,... {{OEIS|A003401}}.

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

{{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^e (mod n)}}.

It is decrypted by computing {{math|t {{=}} S^d (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}}.

{{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) }} divides {{math|n − 1}}. None are known.Ribenboim, pp. 36–37.

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 > 10²⁰}} and that {{math|ω(n) ≥ 14}}.{{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 }} Further, Hagis showed that if 3 divides {{mvar|n}} then {{math|n > 10^1937042}} and {{math|ω(n) ≥ 298848}}.{{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 }}Guy (2004) p.142

{{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 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.

The Riemann hypothesis is true if and only if the inequality

:

is true for all {{math|n ≥ p₁₂₀₅₆₉#}} where {{mvar|γ}} is Euler's constant and {{math|p₁₂₀₅₆₉#}} is the product of the first {{math|120569}} primes.{{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

• Carmichael function (λ)
• Dedekind psi function (𝜓)
• Divisor function (σ)

• Duffin–Schaeffer conjecture
• Generalizations of Fermat's little theorem
• Highly composite number

• Multiplicative group of integers modulo n
• Ramanujan sum
• Totient summatory function (𝛷)

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==External links==

• {{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}}

Category:Modular arithmetic

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Category:Algebra

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