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 Dirichlet $L$-functions

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§25.15(i) Definitions and Basic Properties

►The notation $L(s,\chi )$ was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series … … ►

§25.15(ii) Zeros

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3: 23.2 Definitions and Periodic Properties

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23.2.5 $$\zeta \left(z\right)=\frac{1}{z}+\sum _{w\in 𝕃\setminus \{0\}}\left(\frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^2}\right),$$

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Defines:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function

Symbols:

$g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $\zeta (z;g_2,g_3)$: Weierstrass zeta function, $\in$: element of, $\setminus$: set subtraction, $𝕃$: lattice and $z$: complex

Keywords:

modular functions, Weierstrass zeta function

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.2.E5

Encodings:

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See also:

Annotations for §23.2(ii), §23.2 and Ch.23

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23.2.6 $$\sigma \left(z\right)=z\prod _{w\in 𝕃\setminus \{0\}}\left(\left(1-\frac{z}{w}\right)\exp \left(\frac{z}{w}+\frac{z^2}{2w^2}\right)\right).$$

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Defines:

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function

Symbols:

$g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $\sigma (z;g_2,g_3)$: Weierstrass sigma function, $\in$: element of, $\exp z$: exponential function, $\setminus$: set subtraction, $𝕃$: lattice and $z$: complex

Keywords:

modular functions, Weierstrass sigma function

Referenced by:

§23.2(iii)

Permalink:

http://dlmf.nist.gov/23.2.E6

Encodings:

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See also:

Annotations for §23.2(ii), §23.2 and Ch.23

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4: 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 $G(n,\chi )$ associated with a Dirichlet character $\chi (modk)$. It is defined by the relation … ►For any Dirichlet character $\chi (modk)$, $G(n,\chi )$ is separable for $n$ if $\left(n,k\right)=1$, and is separable for every $n$ if and only if $G(n,\chi )=0$ whenever $\left(n,k\right)>1$. … ►The finite Fourier expansion of a primitive Dirichlet character $\chi (modk)$ has the form …

5: 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 ►

27.5.1 $$h(n)=\sum _{d|n}f(d)g\left(\frac{n}{d}\right),$$

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Symbols:

$d$: positive integer, $n$: positive integer, $f$: function, $g$: function and $h$: function

Permalink:

http://dlmf.nist.gov/27.5.E1

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Annotations for §27.5 and Ch.27

►called the Dirichlet product (or convolution) of $f$ and $g$. The set of all number-theoretic functions $f$ with $f(1)\ne 0$ forms an abelian group under Dirichlet multiplication, with the function $\lfloor 1/n\rfloor$ in (27.2.5) as identity element; see Apostol (1976, p. 129). … ►

27.5.6 $$G(x)=\sum _{n\le x}F\left(\frac{x}{n}\right)⟺F(x)=\sum _{n\le x}\mu \left(n\right)G\left(\frac{x}{n}\right),$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $n$: positive integer, $x$: real number, $F(s)$: Dirichlet series and $G(s)$: Dirichlet series

A&S Ref:

24.3.1 II.C

Permalink:

http://dlmf.nist.gov/27.5.E6

Encodings:

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See also:

Annotations for §27.5 and Ch.27

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6: 25.1 Special Notation

… ►The main related functions are the Hurwitz zeta function $\zeta (s,a)$, the dilogarithm ${\mathrm{Li}}_2\left(z\right)$, the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (also known as Jonquière’s function $\varphi (z,s)$), Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$, and the Dirichlet $L$-functions $L(s,\chi )$.

7: 25.19 Tables

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Cloutman (1989) tabulates $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, 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.

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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 $\zeta (s,a)$, 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.

8: 27.4 Euler Products and Dirichlet Series

§27.4 Euler Products and Dirichlet Series

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27.4.4 $$F(s)=\sum _{n=1}^{\mathrm{\infty }}f(n)n^{-s},$$

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Symbols:

$n$: positive integer, $f$: multiplicative function and $F(s)$: generating function

Permalink:

http://dlmf.nist.gov/27.4.E4

Encodings:

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See also:

Annotations for §27.4 and Ch.27

►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. …

9: 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)). …

10: 25.21 Software

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§25.21(ix) Dirichlet $L$-series

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