Dirichlet \(L\)-functions refresher¶

Dirichlet characters package congruence information; Dirichlet \(L\)-functions turn that into analytic objects whose zeros control prime distribution in residue classes.

Core definitions¶

Let \(\chi\) be a Dirichlet character modulo \(q\). The Dirichlet \(L\)-function is

\[ L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}, \qquad \Re(s) > 1. \]

For \(\Re(s)>1\), it has an Euler product:

\[ L(s,\chi) = \prod_{p \nmid q}\left(1-\frac{\chi(p)}{p^s}\right)^{-1}. \]

For the principal character:

\[ L(s,\chi_0) = \zeta(s)\prod_{p\mid q}\left(1-p^{-s}\right). \]

The decisive analytic fact (used in Dirichlet’s theorem) is:

\[ L(1,\chi) \ne 0 \quad \text{for every nonprincipal character } \chi. \]

What experiments usually visualize or measure¶

Convergence of partial sums \(\sum_{n\le N}\chi(n)n^{-s}\) for various \(s\).
Convergence of partial Euler products over primes.
Sensitivity near \(s=1\) (slow convergence) and how to stabilize it.

Practical numerical caveats¶

Near \(s=1\), both series and Euler products converge painfully slowly.
For Euler products, compute via logs:

\[ \log L(s,\chi) \approx -\sum_{p\le P}\log\left(1-\chi(p)p^{-s}\right) \]

to reduce catastrophic multiplication error.
If you compare characters, always keep the same prime cutoff \(P\) to make plots comparable.

References¶

See References.

[Davenport, 2000, Lejeune Dirichlet, 1837, Serre, 1973, Washington, 1997]

Experiments in this repository¶

E110 — Dirichlet L-series partial sums at s=1 and s=1/2 (principal vs nonprincipal).
E111 — Euler product vs Dirichlet series truncations for L(s,χ): error vs cutoff.