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''Euler totient function, Euler totient''

Another frequently used named for the [[Euler function|Euler function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t1100401.png" />, which counts the natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t1100402.png" /> that are relatively prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t1100403.png" />.

The Carmichael conjecture on the Euler totient function states that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t1100404.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t1100405.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t1100406.png" />; i.e. no value of the Euler function is assumed once. This has now been verified for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t1100407.png" />, [[#References|[a1]]].

A natural generalization of the Euler totient function is the Jordan totient function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t1100408.png" />, which counts the number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t1100409.png" />-tuples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004011.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004012.png" />. Clearly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004013.png" />.

One has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004014.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004015.png" /> runs over the prime numbers dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004016.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004017.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004018.png" /> is the [[Möbius function|Möbius function]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004019.png" /> runs over all divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004020.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004021.png" /> these formulas reduce to the well-known formulas for the Euler function.

The Lehmer problem on the Euler totient function asks for the solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004023.png" />, [[#References|[a2]]]. For some results on this still (1996) largely open problem, see [[#References|[a3]]] and the references therein. The corresponding problem for the Jordan totient function (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004024.png" />) is easy, [[#References|[a4]]]: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004026.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004027.png" /> is a prime number. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004028.png" /> is a prime number, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004029.png" />.

For much more information on the Euler totient function, the Jordan totient function and various other generalizations, see [[#References|[a5]]], [[#References|[a6]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Schlafly,   S. Wagon,   "Carmichael's conjecture on the Euler function is valid below <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004030.png" />"  ''Math. Comp.'' , '''63'''  (1994)  pp. 415–419</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.H. Lehmer,   "On Euler's totient function"  ''Bull. Amer. Math. Soc.'' , '''38'''  (1932)  pp. 745–751</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V. Siva Rama Prasad,   M. Rangamma,   "On composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004031.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110040/t11004032.png" />"  ''Nieuw Archief voor Wiskunde (4)'' , '''5'''  (1987)  pp. 77–83</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.V. Subbarao,   V. Siva Rama Prasad,   "Some analogues of a Lehmer problem on the totient function"  ''Rocky Mount. J. Math.'' , '''15'''  (1985)  pp. 609–620</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Sivamarakrishnan,   "The many facets of Euler's totient II: generalizations and analogues"  ''Nieuw Archief Wiskunde (4)'' , '''8'''  (1990)  pp. 169–188</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Sivamarakrishnan,   "The many facets of Euler's totient I"  ''Nieuw Archief Wiskunde (4)'' , '''4'''  (1986)  pp. 175–190</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L.E. Dickson,   "History of the theory of numbers" , '''I: Divisibility and primality''' , Chelsea, reprint  (1971)  pp. Chapt. V; 113–155</TD></TR></table>