# Dirichlet L-functions

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##### 1: 27.8 Dirichlet Characters
###### §27.8 Dirichlet Characters
In other words, Dirichlet characters (mod $k$) satisfy the four conditions: … If $\chi$ is a character (mod $k$), so is its complex conjugate $\overline{\chi}$. … A divisor $d$ of $k$ is called an induced modulus for $\chi$ if … Every Dirichlet character $\chi$ (mod $k$) is a product …
##### 2: 25.15 Dirichlet $L$-functions
###### §25.15(i) Definitions and Basic Properties
The notation $L\left(s,\chi\right)$ was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series … …
##### 3: 27.10 Periodic Number-Theoretic Functions
Examples are the Dirichlet characters (mod $k$) and the greatest common divisor $\left(n,k\right)$ regarded as a function of $n$. … Another generalization of Ramanujan’s sum is the Gauss sum $G\left(n,\chi\right)$ associated with a Dirichlet character $\chi\pmod{k}$. It is defined by the relation … For any Dirichlet character $\chi\pmod{k}$, $G\left(n,\chi\right)$ is separable for $n$ if $\left(n,k\right)=1$, and is separable for every $n$ if and only if $G\left(n,\chi\right)=0$ whenever $\left(n,k\right)>1$. … The finite Fourier expansion of a primitive Dirichlet character $\chi\pmod{k}$ has the form …
##### 4: 27.5 Inversion Formulas
If a Dirichlet series $F(s)$ generates $f(n)$, and $G(s)$ generates $g(n)$, then the product $F(s)G(s)$ generates called the Dirichlet product (or convolution) of $f$ and $g$. The set of all number-theoretic functions $f$ with $f(1)\neq 0$ forms an abelian group under Dirichlet multiplication, with the function $\left\lfloor 1/n\right\rfloor$ in (27.2.5) as identity element; see Apostol (1976, p. 129). …
27.5.6 $G(x)=\sum_{n\leq x}F\left(\frac{x}{n}\right)\Longleftrightarrow F(x)=\sum_{n% \leq x}\mu\left(n\right)G\left(\frac{x}{n}\right),$
##### 5: 25.1 Special Notation
The main related functions are the Hurwitz zeta function $\zeta\left(s,a\right)$, the dilogarithm $\mathrm{Li}_{2}\left(z\right)$, the polylogarithm $\mathrm{Li}_{s}\left(z\right)$ (also known as Jonquière’s function $\phi\left(z,s\right)$), Lerch’s transcendent $\Phi\left(z,s,a\right)$, and the Dirichlet $L$-functions $L\left(s,\chi\right)$.
##### 6: 25.19 Tables
• Cloutman (1989) tabulates $\Gamma\left(s+1\right)F_{s}(x)$, where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

• Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta\left(s\right)$ for both real and complex $s$. §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta\left(s,a\right)$, and §22.17 lists tables for some Dirichlet $L$-functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

• ##### 7: 27.4 Euler Products and Dirichlet Series
###### §27.4 Euler Products and Dirichlet Series
called Dirichlet series with coefficients $f(n)$. The function $F(s)$ is a generating function, or more precisely, a Dirichlet generating function, for the coefficients. …
##### 8: 14.31 Other Applications
Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). …
##### 9: 23.2 Definitions and Periodic Properties
23.2.5 ${}\zeta\left(z\right)=\frac{1}{z}+\sum_{w\in\mathbb{L}\setminus\{0\}}\left(% \frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^{2}}\right),$
23.2.6 ${}\sigma\left(z\right)=z\prod_{w\in\mathbb{L}\setminus\{0\}}\left(\left(1-% \frac{z}{w}\right)\exp\left(\frac{z}{w}+\frac{z^{2}}{2w^{2}}\right)\right).$