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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 $\overline{\chi }$. … ►Every Dirichlet character $\chi$ (mod $k$) is a product …where ${\chi }_0$ is a character (mod $d$) for some induced modulus $d$ for $\chi$, and ${\chi }_1$ is the principal character (mod $k$). …

2: 25.15 Dirichlet $L$-functions

§25.15 Dirichlet $L$-functions

… ►where $\chi (n)$ is a Dirichlet character $(modk)$ (§27.8). For the principal character ${\chi }_1(modk)$, $L(s,{\chi }_1)$ is analytic everywhere except for a simple pole at $s=1$ with residue $\varphi \left(k\right)/k$, where $\varphi \left(k\right)$ is Euler’s totient function (§27.2). … … ►When $\chi$ is a primitive character (mod $k$) the $L$-functions satisfy the functional equation: …

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 $G(n,\chi )$ 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 …

4: 27.9 Quadratic Characters

§27.9 Quadratic Characters

►For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. …The Legendre symbol $(n|p)$, as a function of $n$, is a Dirichlet character (mod $p$). … ►If an odd integer $P$ has prime factorization $P={\prod }_{r=1}^{\nu \left(n\right)}{p}_r^{a_r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod }_{r=1}^{\nu \left(n\right)}{(n|p_r)}^{a_r}$, with $(n|1)=1$. The Jacobi symbol $(n|P)$ is a Dirichlet character (mod $P$). …

5: 25.19 Tables

… ►

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.

6: 24.16 Generalizations

… ►

§24.16(ii) Character Analogs

►Let $\chi$ be a primitive Dirichlet character $modf$ (see §27.8). Then $f$ is called the conductor of $\chi$. … ►

24.16.11 $$B_{n,\chi }(x)=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_{k,\chi }{x}^{n-k}.$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer, $x$: real or complex, $\chi (a)$: Dirichlet character and $B_{n,\chi }$: Generalized Bernoulli numbers

Permalink:

http://dlmf.nist.gov/24.16.E11

Encodings:

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See also:

Annotations for §24.16(ii), §24.16 and Ch.24

►Let ${\chi }_0$ be the trivial character and ${\chi }_4$ the unique (nontrivial) character with $f=4$; that is, ${\chi }_4(1)=1$, ${\chi }_4(3)=-1$, ${\chi }_4(2)={\chi }_4(4)=0$. …

7: Bibliography D

… ►

Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire $Mx+N$ . Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

Encodings:

BibTeX

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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.

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External Links:

ISSN 0022-314X, Document, MathReview (T. Metsänkylä)

Referenced by:

§24.12(iii).

Encodings:

BibTeX

… ►

Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1837, pp. 45–81 (German).

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Notes:

Reprinted in G. Lejeune Dirichlet’s Werke, Band I, pp. 313–342, Reimer, Berlin, 1889 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.

External Links:

ISBN 0-8284-0225-6, Link, MathReview

Referenced by:

§25.15(i).

Encodings:

BibTeX

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Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1849, pp. 69–83 (German).

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Notes:

Reprinted in G. Lejeune Dirichlet’s Werke, Band II, pp. 49–66, Reimer, Berlin, 1897 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.

External Links:

ISBN 0-8284-0225-6, Link, MathReview

Referenced by:

§27.11.

Encodings:

BibTeX

… ►

Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

…

8: 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

Encodings:

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See also:

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

…

9: 27.3 Multiplicative Properties

… ►Examples are $\lfloor 1/n\rfloor$ and $\lambda \left(n\right)$, and the Dirichlet characters, defined in §27.8. …

10: 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 $L(s,\chi )$.