Dirichlet L-Functions

Dirichlet L-functions generalize the Riemann zeta function by twisting with a character. Their analytic properties -- meromorphic continuation, functional equation, and non-vanishing -- underpin the equidistribution of primes in arithmetic progressions.

Definition and Analytic Continuation

Definition5.4Dirichlet L-Function

For a Dirichlet character $\chi$ mod $q$ , the Dirichlet L-function is $L(s, \chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s} = \prod_{p} \left(1 - \frac{\chi(p)}{p^s}\right)^{-1}$ for $\mathrm{Re}(s) > 1$ . For $\chi \neq \chi_0$ , the series converges conditionally for $\mathrm{Re}(s) > 0$ (by Dirichlet's test, since partial sums of $\chi(n)$ are bounded). For $\chi = \chi_0$ : $L(s, \chi_0) = \zeta(s)\prod_{p|q}(1 - p^{-s})$ , which has a simple pole at $s = 1$ .

Definition5.5Analytic Continuation and Functional Equation

For a primitive character $\chi$ mod $q$ with $\chi(-1) = (-1)^a$ ( $a \in \{0,1\}$ ), define the completed L-function $\Lambda(s, \chi) = \left(\frac{q}{\pi}\right)^{(s+a)/2} \Gamma\!\left(\frac{s+a}{2}\right) L(s, \chi)$ . Then $\Lambda(s,\chi)$ extends to an entire function (for $\chi \neq \chi_0$ ) satisfying the functional equation: $\Lambda(s, \chi) = \frac{\tau(\chi)}{i^a \sqrt{q}} \Lambda(1 - s, \overline{\chi})$

Non-Vanishing and Zeros

ExampleZeros of L-Functions

Like the Riemann zeta function, $L(s, \chi)$ has trivial zeros at negative integers (determined by the Gamma factor) and nontrivial zeros in the critical strip $0 < \mathrm{Re}(s) < 1$ . The Generalized Riemann Hypothesis (GRH) asserts that all nontrivial zeros of $L(s, \chi)$ have $\mathrm{Re}(s) = 1/2$ . The zero-free region $\sigma > 1 - c/\log(q(|t|+2))$ (with possible Siegel zeros) suffices for most applications.

RemarkSiegel Zeros

A Siegel zero is a potential real zero $\beta$ of $L(s, \chi)$ for a real character $\chi$ with $\beta$ close to 1. The Siegel-Walfisz theorem states that $\beta < 1 - c(\varepsilon)q^{-\varepsilon}$ for any $\varepsilon > 0$ , but $c(\varepsilon)$ is ineffective. The existence of Siegel zeros is the main obstacle to proving an effective version of the prime number theorem in arithmetic progressions. Ruling out Siegel zeros would have profound consequences, including effective bounds on the least prime in an arithmetic progression.