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Fixed a typo in the 9th formula. The correct one was taken form the german version, https://de.wikipedia.org/wiki/Eulersche_Phi-Funktion |
Undid revision 1306939706 by Mikhael Username (talk) since, in this context, log(x) is more common than ln(x). Adding the template {{log(x)}} |
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{{Short description|Number of integers coprime to and
{{hatnote group|
{{Redirect|φ(n)||Phi}}
{{distinguish|Euler function}}
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{{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
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[[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 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|''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 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 {{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}}.
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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
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====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}
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&=& 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===
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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
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:
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=\frac3{\pi^2}n^2+O\left(n(\log n)^\frac23(\log\log n)^\frac43\right)</math> (<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>)</li>
<li><math>\sum_{k=1}^n
<li><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> <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></li>▼
<li><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> <ref name=Sita />▼
▲<li><math>\sum_{k=1}^n\frac
▲<li><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> <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></li>
<li><math>\sum_
</ul>
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===Divisibility by any fixed positive integer===
The following property, which is part of the « folklore » (i.e., apparently unpublished as a specific result:<ref> {{citation | last = Pollack | first = P. | title = Two problems on the distribution of
| url = | doi-access = free | arxiv = 2303.14043 }}</ref> 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 <math>\varphi(n)</math> in the arithmetic progressions modulo <math>q</math> for any integer <math>q>1</math>.
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is dense in the interval (0,1).
==Totient
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>
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:<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.
If counted accordingly to multiplicity, the number of totient numbers up to a given limit {{mvar|x}} is
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===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'' ≥ ''p''<sub>120569</sub>#}} where {{mvar|γ}} is [[Euler's constant]] and {{math|''p''<sub>120569</sub>#}} 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>
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*[[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]]
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{{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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| year = 1979
| isbn = 978-0-19-853171-5}}
*{{citation
| last = Liu | first = H.-Q.
| journal = Proc. Roy. Soc. Edinburgh Sect. A
| title = On Euler's function
| volume = 146
| issue = 4
| year = 2016 | pages = 769–775
| doi = 10.1017/S0308210515000682
}}.
* {{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 }}
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