Editing Euler's totient function (section)

==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 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 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.