Editing Arithmetic function (section)

=== Miscellaneous ===
Let ''m'' and ''n'' be distinct, odd, and positive. Then the Jacobi symbol satisfies the law of [[quadratic reciprocity]]:
<math display="block"> \left(\frac{m}{n}\right) \left(\frac{n}{m}\right) = (-1)^{(m-1)(n-1)/4}.</math>

Let ''D''(''n'') be the arithmetic derivative. Then the logarithmic derivative <math display="block">\frac{D(n)}{n} = \sum_{\stackrel{p\mid n}{p\text{ prime}}} \frac {v_{p}(n)} {p}.</math> See ''[[Arithmetic derivative]]'' for details.

Let ''λ''(''n'') be Liouville's function. Then
: <math>|\lambda(n)|\mu(n) =\lambda(n)|\mu(n)| = \mu(n),</math> &nbsp; &nbsp; and
: <math>\lambda(n)\mu(n) = |\mu(n)| =\mu^2(n).</math> &nbsp; &nbsp;

Let ''λ''(''n'') be Carmichael's function. Then
: <math>\lambda(n)\mid \phi(n).</math>  &nbsp; &nbsp;  Further,
: <math>\lambda(n)= \phi(n) \text{ if and only if }n=\begin{cases}
1,2, 4;\\
3,5,7,9,11, \ldots \text{ (that is, } p^k \text{, where }p\text{ is an odd prime)};\\
6,10,14,18,\ldots \text{ (that is, } 2p^k\text{, where }p\text{ is an odd prime)}.
\end{cases}</math> 
See [[Multiplicative group of integers modulo n]] and [[Primitive root modulo n]].
&nbsp;
: <math>2^{\omega(n)} \le d(n) \le 2^{\Omega(n)}.</math> &nbsp; &nbsp;<ref>Hardy ''Ramanujan'', eq. 3.10.3</ref><ref>Hardy & Wright, § 22.13</ref>
: <math>\frac{6}{\pi^2}<\frac{\phi(n)\sigma(n)}{n^2} < 1.</math> &nbsp; &nbsp;<ref>Hardy & Wright, Thm. 329</ref>
: <math>\begin{align}
c_q(n)
&=\frac{\mu\left(\frac{q}{\gcd(q, n)}\right)}{\phi\left(\frac{q}{\gcd(q, n)}\right)}\phi(q)\\
&=\sum_{\delta\mid \gcd(q,n)}\mu\left(\frac{q}{\delta}\right)\delta.
\end{align}</math> &nbsp; &nbsp;<ref>Hardy & Wright, Thms. 271, 272</ref> &nbsp; &nbsp; Note that &nbsp;<math>\phi(q) = \sum_{\delta\mid q}\mu\left(\frac{q}{\delta}\right)\delta.</math> &nbsp; &nbsp;<ref>Hardy & Wright, eq. 16.3.1</ref>
: <math>c_q(1) = \mu(q).</math>
: <math>c_q(q) = \phi(q).</math>
: <math>\sum_{\delta\mid n}d^{3}(\delta) = \left(\sum_{\delta\mid n}d(\delta)\right)^2.</math> &nbsp; &nbsp;<ref>Ramanujan, ''Some Formulæ in the Analytic Theory of Numbers'', eq. (C); ''Papers'' p. 133. A footnote says that Hardy told Ramanujan it also appears in an 1857 paper by Liouville.</ref> &nbsp; Compare this with {{math|1=1<sup>3</sup> + 2<sup>3</sup> + 3<sup>3</sup> + ... + ''n''<sup>3</sup> = (1 + 2 + 3 + ... + ''n'')<sup>2</sup>}}
: <math>d(uv) = \sum_{\delta\mid \gcd(u,v)}\mu(\delta)d\left(\frac{u}{\delta}\right)d\left(\frac{v}{\delta}\right).
</math> &nbsp; &nbsp;<ref>Ramanujan, ''Some Formulæ in the Analytic Theory of Numbers'', eq. (F); ''Papers'' p. 134</ref>
: <math>\sigma_k(u)\sigma_k(v) = \sum_{\delta\mid \gcd(u,v)}\delta^k\sigma_k\left(\frac{uv}{\delta^2}\right).
</math>  &nbsp; &nbsp;<ref>Apostol, ''Modular Functions ...'', ch. 6 eq. 4</ref>
: <math>\tau(u)\tau(v) = \sum_{\delta\mid \gcd(u,v)}\delta^{11}\tau\left(\frac{uv}{\delta^2}\right),
</math>  &nbsp; &nbsp; where ''τ''(''n'') is Ramanujan's function.  &nbsp; &nbsp;<ref>Apostol, ''Modular Functions ...'', ch. 6 eq. 3</ref>