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1: 27.8 Dirichlet Characters

§27.8 Dirichlet Characters

… ►An example is the principal character (mod $k$): … ►Every Dirichlet character $\chi$ (mod $k$) is a product …A character is real if all its values are real. If $k$ is odd, then the real characters (mod $k$) are the principal character and the quadratic characters described in the next section.

2: 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 $n$. … ►where ${\chi }_1$ is the principal character (mod $k$). … ►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 … ►The finite Fourier expansion of a primitive Dirichlet character $\chi (modk)$ has the form …

3: 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 $P$). …

4: 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). … ►where ${\chi }_0$ is a primitive character (mod $d$) for some positive divisor $d$ of $k$ (§27.8). ►When $\chi$ is a primitive character (mod $k$) the $L$-functions satisfy the functional equation: … ►where ${\chi }_1$ is the principal character $(modk)$. …

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

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

6: 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.

7: Guide to Searching the DLMF

… ► ►►►

$	stands for any number of alphanumeric characters
…
?	stands for any zero or one alphanumeric character.
…

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8: 27.3 Multiplicative Properties

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

9: DLMF Project News

error generating summary

10: Bibliography D

… ►

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:

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