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1: 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
…2: 20 Theta Functions
Chapter 20 Theta Functions
…3: 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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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics.
Comput. Phys. Comm. 150 (1), pp. 1–20.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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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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4: 28 Mathieu Functions and Hill’s Equation
Chapter 28 Mathieu Functions and Hill’s Equation
…5: Bibliography G
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Formulas of the Dirichlet-Mehler Type.
In Fractional Calculus and its Applications, B. Ross (Ed.),
Lecture Notes in Math., Vol. 457, pp. 207–215.
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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software 20 (1), pp. 21–62.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Dirichlet convolution of cotangent numbers and relative class number formulas.
Monatsh. Math. 110 (3-4), pp. 231–256.
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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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6: 27.5 Inversion Formulas
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►If a Dirichlet series generates , and generates , then the product
generates
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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).
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27.5.6
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7: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann.
Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).
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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
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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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: Staff
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William P. Reinhardt, University of Washington, Chaps. 20, 22, 23
Hans Volkmer, University of Wisconsin, Milwaukee, Chaps. 29, 30
Peter L. Walker, American University of Sharjah, Chaps. 20, 22, 23
William P. Reinhardt, University of Washington, for Chaps. 20, 22, 23
Peter L. Walker, American University of Sharjah, for Chaps. 20, 22, 23
9: 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 …10: 27.2 Functions
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►Functions in this section derive their properties from the fundamental
theorem of arithmetic, which states that every integer can be represented uniquely as a product of prime powers,
…Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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►It is the special case of the function that counts the number of ways of expressing as the product of factors, with the order of factors taken into account.
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