# Dirichlet L-functions

## Definitions

Symbol: DirichletL $L\!\left(s, \chi\right)$ Dirichlet L-function

## L-series

$L\!\left(s, \chi\right) = \sum_{n=1}^{\infty} \frac{\chi(n)}{{n}^{s}}$
$\frac{1}{L\!\left(s, \chi\right)} = \sum_{n=1}^{\infty} \frac{\mu(n) \chi(n)}{{n}^{s}}$

## Euler product

$L\!\left(s, \chi\right) = \prod_{p} \frac{1}{1 - \chi(p) {p}^{-s}}$
$\frac{1}{L\!\left(s, \chi\right)} = \prod_{p} \left(1 - \frac{\chi(p)}{{p}^{s}}\right)$

## Hurwitz zeta representation

Symbol: HurwitzZeta $\zeta\!\left(s, a\right)$ Hurwitz zeta function
$L\!\left(s, \chi\right) = \frac{1}{{q}^{s}} \sum_{k=1}^{q} \chi(k) \zeta\!\left(s, \frac{k}{q}\right)$
$\zeta\!\left(s, \frac{k}{q}\right) = \frac{{q}^{s}}{\varphi(q)} \sum_{\chi \in G_{q}} \overline{\chi(k)} L\!\left(s, \chi\right)$

## Principal and non-primitive characters

$L\!\left(s, \chi_{1 \, . \, 1}\right) = \zeta\!\left(s\right)$
$L\!\left(s, \chi_{{2}^{n} \, . \, 1}\right) = \left(1 - {2}^{-s}\right) \zeta\!\left(s\right)$
$L\!\left(s, \chi_{q \, . \, 1}\right) = \zeta\!\left(s\right) \prod_{p \mid q} \left(1 - \frac{1}{{p}^{s}}\right)$
$L\!\left(s, \chi\right) = L\!\left(s, {\chi}_{0}\right) \prod_{p \mid q} \left(1 - \frac{{\chi}_{0}\!\left(p\right)}{{p}^{s}}\right)\; \text{ where } {\chi}_{1} = \chi_{q \, . \, 1},\;\chi = {\chi}_{0} {\chi}_{1}$

## Value at 1

Related topic: Stieltjes constants
$L\!\left(1, \chi\right) = \begin{cases} {\tilde \infty}, & \chi = \chi_{q \, . \, 1}\\\lim_{s \to 1} L\!\left(s, \chi\right), & \text{otherwise}\\ \end{cases}$
$L\!\left(1, \chi\right) \ne 0$
$L\!\left(1, \chi\right) \notin \overline{\mathbb{Q}}$
$\lim_{s \to 1} \left(s - 1\right) L\!\left(1, \chi_{q \, . \, 1}\right) = \frac{\varphi(q)}{q}$
$L\!\left(1, \chi\right) = -\frac{1}{q} \sum_{k=1}^{q - 1} \chi(k) \psi\!\left(\frac{k}{q}\right)$
$L\!\left(1, \chi_{3 \, . \, 2}\right) = \frac{\pi}{\sqrt{27}}$
$L\!\left(1, \chi_{4 \, . \, 3}\right) = \frac{\pi}{4}$
$L\!\left(1, \chi_{5 \, . \, 4}\right) = \frac{2 \log(\varphi)}{\sqrt{5}}$

## Value at 0

$L\!\left(0, \chi_{q \, . \, 1}\right) = \begin{cases} -\frac{1}{2}, & q = 1\\0, & \text{otherwise}\\ \end{cases}$
$L\!\left(0, \chi\right) = -\frac{1}{q} \sum_{k=1}^{q} k \chi(k)$
$\left(\chi(-1) = 1\right) \implies \left(L\!\left(0, \chi\right) = 0\right)$

## Values at negative integers

Symbol: GeneralizedBernoulliB $B_{n,\chi}$ Generalized Bernoulli number
$B_{n,\chi} = \sum_{a=1}^{q} \chi(a) \sum_{k=0}^{n} {n \choose k} B_{k} {a}^{n - k} {q}^{k - 1}$
$B_{n,\chi} = {q}^{n - 1} \sum_{a=1}^{q} \chi(a) B_{n}\!\left(\frac{a}{q}\right)$
$B_{0,\chi} = \begin{cases} \frac{\varphi(q)}{q}, & \chi = \chi_{q \, . \, 1}\\0, & \text{otherwise}\\ \end{cases}$
$\sum_{a=1}^{q} \chi(a) \frac{z {e}^{a z}}{{e}^{q z} - 1} = \sum_{n=0}^{\infty} B_{n,\chi} \frac{{z}^{n}}{n !}$
$L\!\left(-n, \chi\right) = -\frac{B_{n + 1,\chi}}{n + 1}$

## Zeros

### Nontrivial zeros

Symbol: DirichletLZero $\rho_{n,\chi}$ Nontrivial zero of Dirichlet L-function
Symbol: GeneralizedRiemannHypothesis $\operatorname{GRH}$ Generalized Riemann hypothesis
$\left(\operatorname{GRH}\right) \iff \left(\operatorname{Re}\!\left(\rho_{n,\chi}\right) = \frac{1}{2} \;\text{ for all } \left(q, \chi, n\right) \text{ with } q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \setminus \left\{0\right\}\right)$
$0 < \operatorname{Re}\!\left(\rho_{n,\chi}\right) < 1$
$\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,0 < \operatorname{Re}(s) < 1} L\!\left(s, \chi\right) = \left\{ \rho_{n,\chi} : n \in \mathbb{Z} \setminus \left\{0\right\} \right\}$
$\operatorname{Re}\!\left(\rho_{n,\chi}\right) = \frac{1}{2}$

### Trivial zeros

$\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C}} L\!\left(s, \chi\right) = \left(\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,\operatorname{Re}(s) \le 0} L\!\left(s, \chi\right)\right) \cup \left\{ \rho_{n,\chi} : n \in \mathbb{Z} \setminus \left\{0\right\} \right\}$
$\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,\operatorname{Re}(s) \le 0} L\!\left(s, \chi\right) = \begin{cases} \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\}, & q = 1\\\left\{ -2 n : n \in \mathbb{Z}_{\ge 0} \right\}, & \chi(-1) = 1 \;\mathbin{\operatorname{and}}\; q \ne 1\\\left\{ -2 n - 1 : n \in \mathbb{Z}_{\ge 0} \right\}, & \chi(-1) = -1\\ \end{cases}$

## Conjugate symmetry

$L\!\left(s, \overline{\chi}\right) = \overline{L\!\left(\overline{s}, \chi\right)}$
$L\!\left(\overline{s}, \chi\right) = \overline{L\!\left(s, \overline{\chi}\right)}$
$L\!\left(\overline{s}, \overline{\chi}\right) = \overline{L\!\left(s, \chi\right)}$

## Functional equation

Symbol: DirichletLambda $\Lambda\!\left(s, \chi\right)$ Completed Dirichlet L-function
$\Lambda\!\left(s, \chi\right) = \beta {\left(\frac{q}{\pi}\right)}^{\left( s + a \right) / 2} \Gamma\!\left(\frac{s + a}{2}\right) L\!\left(s, \chi\right)\; \text{ where } a = \frac{1 - \chi(-1)}{2},\;\beta = \begin{cases} s \left(s - 1\right), & q = 1\\1, & \text{otherwise}\\ \end{cases}$
Symbol: GaussSum $G_{q}\!\left(\chi\right)$ Gauss sum
$G_{q}\!\left(\chi\right) = \sum_{n=1}^{q} \chi(n) {e}^{2 \pi i n / q}$
$\left|G_{q}\!\left(\chi\right)\right| = \sqrt{q}$
$\Lambda\!\left(s, \chi\right) = \varepsilon \Lambda\!\left(1 - s, \overline{\chi}\right)\; \text{ where } a = \frac{1 - \chi(-1)}{2},\;\varepsilon = \frac{G_{q}\!\left(\chi\right)}{{i}^{a} \sqrt{q}}$

## Analytic properties

$\operatorname{BranchCuts}\!\left(L\!\left(s, \chi\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{EssentialSingularities}\!\left(L\!\left(s, \chi\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}$
$\mathop{\operatorname{poles}\,}\limits_{s \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} L\!\left(s, \chi\right) = \begin{cases} \left\{1\right\}, & \chi = \chi_{q \, . \, 1}\\\left\{\right\}, & \text{otherwise}\\ \end{cases}$
$L\!\left(s, \chi\right) \text{ is holomorphic on } s \in \begin{cases} \mathbb{C} \setminus \left\{1\right\}, & \chi = \chi_{q \, . \, 1}\\\mathbb{C}, & \text{otherwise}\\ \end{cases}$

## Approximations

$\left|L\!\left(s, \chi\right) - \sum_{k=1}^{N - 1} \frac{\chi(k)}{{k}^{s}}\right| \le \zeta\!\left(\operatorname{Re}(s), N\right)$
$\left|\frac{1}{L\!\left(s, \chi\right)} - \prod_{p < N} \left(1 - \frac{\chi(p)}{{p}^{s}}\right)\right| \le \zeta\!\left(\operatorname{Re}(s), N\right)$

## Bounds and inequalities

$\left|L\!\left(s, \chi\right)\right| \le \zeta\!\left(\operatorname{Re}(s)\right)$
$\left|L\!\left(s, \chi\right)\right| \le {\left(\frac{q \left|1 + s\right|}{2 \pi}\right)}^{\left( 1 + \eta - \operatorname{Re}(s) \right) / 2} \zeta\!\left(1 + \eta\right)$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC