Dirichlet%20L-functions
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1: 27.8 Dirichlet Characters
§27.8 Dirichlet Characters
… ►In other words, Dirichlet characters (mod ) satisfy the four conditions: … ►If is a character (mod ), so is its complex conjugate . … ►A divisor of is called an induced modulus for if … ►Every Dirichlet character (mod ) is a product …2: 25.15 Dirichlet -functions
§25.15 Dirichlet -functions
►§25.15(i) Definitions and Basic Properties
►The notation 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 ) and the greatest common divisor regarded as a function of .
…
►Another generalization of Ramanujan’s sum is the Gauss sum
associated with a Dirichlet character .
It is defined by the relation
…
►For any Dirichlet character , is separable for if , and is separable for every if and only if whenever .
…
►The finite Fourier expansion of a primitive Dirichlet character has the form
…
4: 27.5 Inversion Formulas
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►If a Dirichlet series generates , and generates , then the product generates
►
27.5.1
►called the Dirichlet product (or convolution) of and .
The set of all number-theoretic functions
with forms an abelian group under Dirichlet multiplication, with the function
in (27.2.5) as identity element; see Apostol (1976, p. 129).
…
►
27.5.6
…
5: 25.1 Special Notation
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►The main related functions are the Hurwitz zeta function
, the dilogarithm , the polylogarithm (also known as Jonquière’s function
), Lerch’s transcendent , and the Dirichlet
-functions
.
6: 25.19 Tables
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•
►
•
Fletcher et al. (1962, §22.1) lists many sources for earlier tables of for both real and complex . §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of , and §22.17 lists tables for some Dirichlet -functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.
7: Bibliography D
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►
Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
.
Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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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).
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Über die Bestimmung der mittleren Werthe in der Zahlentheorie.
Abhandlungen der Königlich Preussischen Akademie der
Wissenschaften von 1849, pp. 69–83 (German).
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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8: 27.4 Euler Products and Dirichlet Series
§27.4 Euler Products and Dirichlet Series
… ►
27.4.4
►called Dirichlet series with coefficients .
The function
is a generating function, or more precisely, a Dirichlet generating
function, for the coefficients.
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9: 14.31 Other Applications
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►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)).
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10: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Dirichlet series related to the Riemann zeta function.
J. Number Theory 19 (1), pp. 85–102.
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Note on the trivial zeros of Dirichlet
-functions.
Proc. Amer. Math. Soc. 94 (1), pp. 29–30.
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Modular Functions and Dirichlet Series in Number Theory.
2nd edition, Graduate Texts in Mathematics, Vol. 41, Springer-Verlag, New York.
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