Editing Euler's totient function (section)

==Other formulae==
<ul>
<li><math>a\mid b \implies \varphi(a)\mid\varphi(b)</math></li>
<li><math> m \mid \varphi(a^m-1)</math></li>
<li><math>\varphi(mn) = \varphi(m)\varphi(n)\cdot\frac{d}{\varphi(d)} \quad\text{where }d = \operatorname{gcd}(m,n)</math>
<p>In particular:</p>
*<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></li>
<li><math>\varphi(\operatorname{lcm}(m,n))\cdot\varphi(\operatorname{gcd}(m,n)) = \varphi(m)\cdot\varphi(n)</math>
<p>Compare this to the formula <math display=inline>\operatorname{lcm}(m,n)\cdot \operatorname{gcd}(m,n) = m \cdot n</math>  (see [[least common multiple]]).</p>
</li>
<li>{{math|''φ''(''n'')}} is even for {{math|''n'' ≥ 3}}. <p>Moreover, if {{mvar|n}} has {{mvar|r}} distinct odd prime factors, {{math|2<sup>''r''</sup> {{!}} ''φ''(''n'')}}</p></li>
<li> 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)}}.</li>
<li><math>\frac{\varphi(n)}{n}=\frac{\varphi(\operatorname{rad}(n))}{\operatorname{rad}(n)}</math>
<p>where {{math|rad(''n'')}} is the [[radical of an integer|radical of {{mvar|n}}]] (the product of all distinct primes dividing {{mvar|[[radical of an integer|n]]}}).</p></li>
<li><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></li>
<li><math>\sum_{1\le k\le n-1 \atop gcd(k,n)=1}\!\!k = \tfrac12 n\varphi(n) \quad \text{for }n>1</math></li>
<li><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 | 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\varphi(k) =\frac3{\pi^2}n^2+O\left(n(\log n)^\frac23(\log\log n)^\frac13\right) </math> [Liu (2016)] </li>

<li><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" /></li>
<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>&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 |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>&nbsp;<ref name="Sita" /><p>(where {{mvar|γ}} is the [[Euler–Mascheroni constant]]).</p></li>
</ul>

===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 {{math|[[Divisor function|''d''(''n'') {{=}} ''σ''<sub>0</sub>(''n'')]]}} is the number of divisors of {{mvar|n}}.

===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 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 }}</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>. 

* For every fixed positive integer <math>q</math>, the relation <math>q|\varphi(n)</math> holds for almost all <math>n</math>, meaning for all but <math>o(x)</math> values of <math>n\le x</math> as <math>x\rightarrow\infty</math>.

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