Dirichlet character

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1: 27.8 Dirichlet Characters

§27.8 Dirichlet Characters

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27.8.2 $$\chi \left(mn\right)=\chi \left(m\right)\chi \left(n\right),$$ $m,n=1,2,\mathrm{\dots }$,

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $k$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E2

Encodings:

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Annotations for §27.8 and Ch.27

… ►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: 27.10 Periodic Number-Theoretic Functions

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27.10.4 $$c_k\left(n\right)=\sum _{m=1}^k{\chi }_1\left(m\right){\mathrm{e}}^{2\pi \mathrm{i}mn/k},$$

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Defines:

$c_k\left(n\right)$: Ramanujan’s sum

Symbols:

$\chi \left(n\right)$: Dirichlet character, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E4

Encodings:

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Annotations for §27.10 and Ch.27

… ►It is defined by the relation … ►

27.10.10 $$G(n,\chi )=\overline{\chi }(n)G(1,\chi ).$$

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $G(n,\chi )$: Gauss sum, $\overline{z}$: complex conjugate, $n$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E10

Encodings:

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See also:

Annotations for §27.10 and Ch.27

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27.10.11 $${|G(1,\chi )|}^2=k.$$

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $G(n,\chi )$: Gauss sum, $k$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E11

Encodings:

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Annotations for §27.10 and Ch.27

… ►The finite Fourier expansion of a primitive Dirichlet character $\chi (modk)$ has the form …

3: 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). … ►

25.15.7 $$L(-2n,\chi )=0\text{ if }\chi (-1)=1,$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $n$: nonnegative integer and $\chi (n)$: Dirichlet character

Keywords:

zeros

Source:

Apostol (1985b, p. 29)

Permalink:

http://dlmf.nist.gov/25.15.E7

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Annotations for §25.15(ii), §25.15 and Ch.25

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25.15.8 $$L(-2n-1,\chi )=0\text{ if }\chi (-1)=-1,$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $n$: nonnegative integer and $\chi (n)$: Dirichlet character

Keywords:

zeros

Source:

Apostol (1985b, p. 29)

Permalink:

http://dlmf.nist.gov/25.15.E8

Encodings:

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See also:

Annotations for §25.15(ii), §25.15 and Ch.25

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25.15.9 $$L(1,\chi )\ne 0\text{ if }\chi \ne {\chi }_1,$$

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function and $\chi (n)$: Dirichlet character

Keywords:

zeros

Source:

Apostol (1976, p. 149)

Permalink:

http://dlmf.nist.gov/25.15.E9

Encodings:

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See also:

Annotations for §25.15(ii), §25.15 and Ch.25

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4: 27.9 Quadratic Characters

§27.9 Quadratic Characters

… ►The Legendre symbol $(n|p)$, as a function of $n$, is a Dirichlet character (mod $p$). … ►The Jacobi symbol $(n|P)$ is a Dirichlet character (mod $P$). …

5: 24.16 Generalizations

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§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

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24.16.12 $$B_n\left(x\right)=B_{n,{\chi }_0}(x-1),$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $n$: integer, $x$: real or complex, $\chi (a)$: Dirichlet character and $B_{n,\chi }$: Generalized Bernoulli numbers

Referenced by:

§24.16(ii)

Permalink:

http://dlmf.nist.gov/24.16.E12

Encodings:

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See also:

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

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6: 27.3 Multiplicative Properties

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

7: 25.19 Tables

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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: 25.11 Hurwitz Zeta Function

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25.11.36Removed because it is just (25.15.1) combined with (25.15.3).

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function and $k$: nonnegative integer

Referenced by:

§25.11(x), Erratum (V1.0.23) for Equation (25.11.36), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.11.E36

Removal (effective with 1.1.4):

This equation has been removed because it is just (25.15.1) combined with (25.15.3).

Clarification (effective with 1.0.23):

The Dirichlet $L$-function was inserted on the left-hand side. The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

See also:

Annotations for §25.11(x), §25.11 and Ch.25

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9: Bibliography D

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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

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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

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