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Dirichlet L-functions

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1: 27.8 Dirichlet Characters
§27.8 Dirichlet Characters
In other words, Dirichlet characters (mod k ) satisfy the four conditions: … If χ is a character (mod k ), so is its complex conjugate χ ¯ . … A divisor d of k is called an induced modulus for χ if … Every Dirichlet character χ (mod k ) is a product …
2: 25.15 Dirichlet L -functions
§25.15 Dirichlet L -functions
§25.15(i) Definitions and Basic Properties
The notation L ( s , χ ) was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series … …
§25.15(ii) Zeros
3: 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 . … 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 … For any Dirichlet character χ ( mod k ) , G ( n , χ ) is separable for n if ( n , k ) = 1 , and is separable for every n if and only if G ( n , χ ) = 0 whenever ( n , k ) > 1 . … The finite Fourier expansion of a primitive Dirichlet character χ ( mod k ) has the form …
4: 27.5 Inversion Formulas
If a Dirichlet series F ( s ) generates f ( n ) , and G ( s ) generates g ( n ) , then the product F ( s ) G ( s ) generates
27.5.1 h ( n ) = d | n f ( d ) g ( n d ) ,
called the Dirichlet product (or convolution) of f and g . The set of all number-theoretic functions f with f ( 1 ) 0 forms an abelian group under Dirichlet multiplication, with the function 1 / n in (27.2.5) as identity element; see Apostol (1976, p. 129). …
27.5.6 G ( x ) = n x F ( x n ) F ( x ) = n x μ ( n ) G ( x n ) ,
5: 25.1 Special Notation
The main related functions are the Hurwitz zeta function ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .
6: 25.19 Tables
  • Cloutman (1989) tabulates Γ ( s + 1 ) F s ( x ) , where F s ( x ) is the Fermi–Dirac integral (25.12.14), for s = - 1 2 , 1 2 , 3 2 , 5 2 , x = - 5 ( .05 ) 25 , to 12S.

  • 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: 27.4 Euler Products and Dirichlet Series
    §27.4 Euler Products and Dirichlet Series
    27.4.4 F ( s ) = n = 1 f ( n ) n - s ,
    called Dirichlet series with coefficients f ( n ) . The function F ( s ) is a generating function, or more precisely, a Dirichlet generating function, for the coefficients. …
    8: 14.31 Other Applications
    Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). …
    9: 23.2 Definitions and Periodic Properties
    23.2.5 ζ ( z ) = 1 z + w 𝕃 { 0 } ( 1 z - w + 1 w + z w 2 ) ,
    23.2.6 σ ( z ) = z w 𝕃 { 0 } ( ( 1 - z w ) exp ( z w + z 2 2 w 2 ) ) .
    10: 25.21 Software
    §25.21(ix) Dirichlet L -series