# characters

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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 $G\left(n,\chi\right)$ associated with a Dirichlet character $\chi\pmod{k}$. It is defined by the relation … The finite Fourier expansion of a primitive Dirichlet character $\chi\pmod{k}$ has the form …
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 $P=\prod_{r=1}^{\nu\left(n\right)}p^{a_{r}}_{r}$, 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 $\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). … 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 $\pmod{k}$. …
##### 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 $B_{n,\chi}(x)=\sum_{k=0}^{n}{n\choose k}B_{k,\chi}x^{n-k}.$
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
• 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.

• ##### 7: Guide to Searching the DLMF
\$ stands for any number of alphanumeric characters … stands for any zero or one alphanumeric character. …
##### 8: 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. …
##### 9: DLMF Project News
error generating summary
##### 10: Bibliography D
• K. Dilcher (1987b) Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.