# Dirichlet character

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##### 1: 27.8 Dirichlet Characters
###### §27.8 DirichletCharacters
27.8.2 $\chi\left(mn\right)=\chi\left(m\right)\chi\left(n\right),$ $m,n=1,2,\dots$,
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
It is defined by the relation …
27.10.10 $G\left(n,\chi\right)=\overline{\chi}(n)G\left(1,\chi\right).$
The finite Fourier expansion of a primitive Dirichlet character $\chi\pmod{k}$ has the form …
##### 3: 25.15 Dirichlet $L$-functions
where $\chi(n)$ is a Dirichlet character $\pmod{k}$27.8). For the principal character $\chi_{1}\pmod{k}$, $L\left(s,\chi_{1}\right)$ is analytic everywhere except for a simple pole at $s=1$ with residue $\phi\left(k\right)/k$, where $\phi\left(k\right)$ is Euler’s totient function (§27.2). …
25.15.7 $L\left(-2n,\chi\right)=0\text{ if }\chi(-1)=1,$ $n=0,1,2,\dots$,
25.15.8 $L\left(-2n-1,\chi\right)=0\text{ if }\chi(-1)=-1,$ $n=0,1,2,\dots$.
25.15.9 $L\left(1,\chi\right)\neq 0\text{ if }\chi\neq\chi_{1},$
##### 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
###### §24.16(ii) Character Analogs
Let $\chi$ be a primitive Dirichlet character $\mod f$ (see §27.8). Then $f$ is called the conductor of $\chi$. …
24.16.11 $B_{n,\chi}(x)=\sum_{k=0}^{n}{n\choose k}B_{k,\chi}x^{n-k}.$
##### 6: 27.3 Multiplicative Properties
Examples are $\left\lfloor 1/n\right\rfloor$ and $\lambda\left(n\right)$, and the Dirichlet characters, defined in §27.8. …
##### 7: 25.19 Tables
• 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.

• ##### 8: 25.11 Hurwitz Zeta Function
25.11.36Removed because it is just (25.15.1) combined with (25.15.3).
##### 9: Bibliography D
• C. de la Vallée Poussin (1896b) 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).
• K. Dilcher (1987b) Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.
• P. G. L. Dirichlet (1837) 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).
• P. G. L. Dirichlet (1849) Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1849, pp. 69–83 (German).