Lagmental Vicfred

Dirichlet Characters Have Finite Fourier Orthogonality by Vicfred

Last updated: Tue 15 August 2017

tags: mathematics, analytic number theory, zeta functions, Dirichlet characters

Summing characters over the unit group isolates the principal character or a residue class. I will separate the object being defined from the consequence being claimed.

Start locally

Dirichlet series \(\sum a_nn^{-s}\) turn multiplicativity into Euler products. The complex variable \(s=\sigma+it\) lets analytic continuation and zero-free regions control arithmetic sums.

$$ \chi:(\mathbf Z/q\mathbf Z)^\times\to\mathbf C^\times $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ \sum_{a\bmod q}\chi(a)\overline{\psi(a)}=\begin{cases}\varphi(q),&\chi=\psi,\\0,&\chi\ne\psi,\end{cases} $$

Compute before generalising

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ \frac1{\varphi(q)}\sum_{\chi\bmod q}\chi(n)\overline{\chi(a)}=\begin{cases}1,&n\equiv a\pmod q,\\0,&n\not\equiv a\pmod q.\end{cases} $$

The global view

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad \chi:(\mathbf Z/q\mathbf Z)^\times\to\mathbf C^\times,\\[5pt] \mathsf{C}\;&:\quad \sum_{a\bmod q}\chi(a)\overline{\psi(a)}=\begin{cases}\varphi(q),&\chi=\psi,\\0,&\chi\ne\psi,\end{cases}. \end{aligned} $$

Edge conditions

An Euler product converges absolutely only in a right half-plane. Formal rearrangement outside that region can destroy the argument.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \sum_{a\bmod q}\chi(a)\overline{\psi(a)}=\begin{cases}\varphi(q),&\chi=\psi,\\0,&\chi\ne\psi,\end{cases} \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Fri 24 August 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.