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1: 27.8 Dirichlet Characters
§27.8 Dirichlet Characters
An example is the principal character (mod k ): … Every Dirichlet character χ (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 ( n , k ) regarded as a function of n . … where χ 1 is the principal character (mod k ). … Another generalization of Ramanujan’s sum is the Gauss sum G ( n , χ ) associated with a Dirichlet character χ ( mod k ) . It is defined by the relation … The finite Fourier expansion of a primitive Dirichlet character χ ( mod k ) 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 = r = 1 ν ( n ) p r a r , then the Jacobi symbol ( n | P ) is defined by ( n | P ) = r = 1 ν ( n ) ( 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 χ ( n ) is a Dirichlet character ( mod k ) 27.8). For the principal character χ 1 ( mod k ) , L ( s , χ 1 ) is analytic everywhere except for a simple pole at s = 1 with residue ϕ ( k ) / k , where ϕ ( k ) is Euler’s totient function (§27.2). … where χ 0 is a primitive character (mod d ) for some positive divisor d of k 27.8). When χ is a primitive character (mod k ) the L -functions satisfy the functional equation: … where χ 1 is the principal character ( mod k ) . …
5: 24.16 Generalizations
§24.16(ii) Character Analogs
Let χ be a primitive Dirichlet character mod f (see §27.8). Then f is called the conductor of χ . …
24.16.11 B n , χ ( x ) = k = 0 n ( n k ) B k , χ x n - k .
Let χ 0 be the trivial character and χ 4 the unique (nontrivial) character with f = 4 ; that is, χ 4 ( 1 ) = 1 , χ 4 ( 3 ) = - 1 , χ 4 ( 2 ) = χ 4 ( 4 ) = 0 . …
6: 25.19 Tables
  • Fletcher et al. (1962, §22.1) lists many sources for earlier tables of ζ ( s ) 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 ζ ( 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.
    8: 27.3 Multiplicative Properties
    Examples are 1 / n and λ ( n ) , 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.