Dirichlet%20characters
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11: 27.4 Euler Products and Dirichlet Series
§27.4 Euler Products and Dirichlet Series
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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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12: 25.11 Hurwitz Zeta Function
13: 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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14: 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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15: 20 Theta Functions
Chapter 20 Theta Functions
…16: 25.21 Software
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§25.21(ix) Dirichlet -series
…17: 27.11 Asymptotic Formulas: Partial Sums
§27.11 Asymptotic Formulas: Partial Sums
… ►For example, Dirichlet (1849) proves that for all , …Dirichlet’s divisor problem (unsolved as of 2022) is to determine the least number such that the error term in (27.11.2) is for all . … ►where , . ►Letting in (27.11.9) or in (27.11.11) we see that there are infinitely many primes if are coprime; this is Dirichlet’s theorem on primes in arithmetic progressions. …18: 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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19: Software Index
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20 Theta Functions | |||||||||||||||||||||||||
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25.21(ix) Dirichlet -series | ✓ | a | ✓ | ✓ | |||||||||||||||||||||
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20: Bibliography M
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Siegel’s modular forms and Dirichlet series.
Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.
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Rational approximations, software and test methods for sine and cosine integrals.
Numer. Algorithms 12 (3-4), pp. 259–272.
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Calculation of the modified Bessel functions of the second kind with complex argument.
Math. Comp. 20 (95), pp. 407–412.
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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems.
Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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