Editing Arithmetic function (section)

=== Class number related ===
[[Peter Gustav Lejeune Dirichlet]] discovered formulas that relate the class number ''h'' of [[quadratic number field]]s to the Jacobi symbol.<ref>Landau, p. 168, credits Gauss as well as Dirichlet</ref>

An integer ''D'' is called a '''fundamental discriminant''' if it is the [[discriminant]] of a quadratic number field. This is equivalent to ''D'' ≠ 1 and either a) ''D'' is [[squarefree]] and ''D'' ≡ 1 (mod 4) or b) ''D'' ≡ 0 (mod 4), ''D''/4 is squarefree, and ''D''/4 ≡ 2 or 3 (mod 4).<ref>Cohen, Def. 5.1.2</ref>

Extend the Jacobi symbol to accept even numbers in the "denominator" by defining the [[Kronecker symbol]]:
<math display="block">\left(\frac{a}{2}\right) = \begin{cases}
\;\;\,0&\text{ if } a \text{ is even}
\\(-1)^{\frac{a^2-1}{8}}&\text{ if }a \text{ is odd. }
\end{cases}</math>

Then if ''D'' < −4 is a fundamental discriminant<ref>Cohen, Corr. 5.3.13</ref><ref>see Edwards, § 9.5 exercises for more complicated formulas.</ref>
<math display="block">\begin{align}
h(D) & = \frac{1}{D} \sum_{r=1}^{|D|}r\left(\frac{D}{r}\right)\\
     & = \frac{1}{2-\left(\tfrac{D}{2}\right)} \sum_{r=1}^{|D|/2}\left(\frac{D}{r}\right).
\end{align}</math>

There is also a formula relating ''r''<sub>3</sub> and ''h''. Again, let ''D'' be a fundamental discriminant, ''D'' < −4. Then<ref>Cohen, Prop 5.3.10</ref>
<math display="block">r_3(|D|) = 12\left(1-\left(\frac{D}{2}\right)\right)h(D).</math>