Editing Arithmetic function (section)

=== Sums of squares ===
For all <math>k \ge 4,\;\;\; r_k(n) > 0.</math> &nbsp; &nbsp; ([[Lagrange's four-square theorem]]).
: <math>
r_2(n) = 4\sum_{d\mid n}\left(\frac{-4}{d}\right),
</math> <ref>Hardy & Wright, Thm. 278</ref>
where the [[Kronecker symbol]] has the values
: <math>
\left(\frac{-4}{n}\right) = 
\begin{cases}
+1&\text{if }n\equiv 1 \pmod 4 \\
-1&\text{if }n\equiv 3 \pmod 4\\
\;\;\;0&\text{if }n\text{ is even}.\\
\end{cases}
</math>

There is a formula for ''r''<sub>3</sub> in the section on [[#Class number related|class numbers]] below.
<math display="block">
r_4(n) =
8 \sum_{\stackrel{d\mid n}{ 4\, \nmid \,d}}d =
8 (2+(-1)^n)\sum_{\stackrel{d\mid n}{ 2\, \nmid \,d}}d =
\begin{cases}
8\sigma(n)&\text{if } n \text{ is odd }\\
24\sigma\left(\frac{n}{2^\nu}\right)&\text{if } n \text{ is even }
\end{cases},
</math>
where {{math|1=''ν'' = ''ν''<sub>2</sub>(''n'')}}. &nbsp; &nbsp;<ref>Hardy & Wright, Thm. 386</ref><ref>Hardy, ''Ramanujan'', eqs 9.1.2, 9.1.3</ref><ref>Koblitz, Ex. III.5.2</ref>
<math display="block">r_6(n) = 16 \sum_{d\mid n} \chi\left(\frac{n}{d}\right)d^2 - 4\sum_{d\mid n} \chi(d)d^2,</math>
where <math> \chi(n) = \left(\frac{-4}{n}\right).</math><ref name="Hardy & Wright, § 20.13">Hardy & Wright, § 20.13</ref>

Define the function {{math|1=''σ''<sub>''k''</sub><sup>*</sup>(''n'')}} as<ref>Hardy, ''Ramanujan'', § 9.7</ref>
<math display="block">\sigma_k^*(n) = (-1)^{n}\sum_{d\mid n}(-1)^d d^k=
\begin{cases}
\sum_{d\mid n} d^k=\sigma_k(n)&\text{if } n \text{ is odd }\\
\sum_{\stackrel{d\mid n}{ 2\, \mid \,d}}d^k -\sum_{\stackrel{d\mid n}{ 2\, \nmid \,d}}d^k&\text{if } n \text{ is even}.
\end{cases}
</math>

That is, if ''n'' is odd, {{math|1=''σ''<sub>''k''</sub><sup>*</sup>(''n'')}} is  the sum of the ''k''th powers of the divisors of ''n'', that is, {{math|1=''σ''<sub>''k''</sub>(''n''),}} and if ''n'' is even it is the sum of the ''k''th  powers of the even divisors of ''n'' minus the sum of the ''k''th  powers of the odd divisors of ''n''.
: <math>r_8(n) = 16\sigma_3^*(n).</math> &nbsp; &nbsp;<ref name="Hardy & Wright, § 20.13" /><ref>Hardy, ''Ramanujan'', § 9.13</ref>

Adopt the convention that Ramanujan's {{math|1=''τ''(''x'') = 0}} if ''x'' is '''not an integer.'''
: <math>
r_{24}(n) = \frac{16}{691}\sigma_{11}^*(n) + \frac{128}{691}\left\{
(-1)^{n-1}259\tau(n)-512\tau\left(\frac{n}{2}\right)\right\}
</math> &nbsp; &nbsp;<ref>Hardy, ''Ramanujan'', § 9.17</ref>