Dirichlet product (or convolution)
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1: 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 .
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2: 25.15 Dirichlet -functions
3: 27.4 Euler Products and Dirichlet Series
§27.4 Euler Products and Dirichlet Series
…4: 27.8 Dirichlet Characters
5: 27.9 Quadratic Characters
§27.9 Quadratic Characters
… ►The Legendre symbol , as a function of , is a Dirichlet character (mod ). … ►If an odd integer has prime factorization , then the Jacobi symbol is defined by , with . The Jacobi symbol is a Dirichlet character (mod ). …6: 27.3 Multiplicative Properties
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27.3.2
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27.3.3
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27.3.4
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27.3.5
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►Examples are and , and the Dirichlet characters, defined in §27.8.
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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
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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Sums of products of Bernoulli numbers.
J. Number Theory 60 (1), pp. 23–41.
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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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Vector coupling coefficients as products of prime factors.
Comput. Phys. Comm. 4 (2), pp. 268–274.
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8: 25.11 Hurwitz Zeta Function
9: Errata
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Section 27.11
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Equation (25.15.10)
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Chapter 19
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Equation (25.11.36)
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References
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25.15.10
The upper-index of the finite sum which originally was , was replaced with since .
Reported by Gergő Nemes on 2021-08-23
Factors inside square roots on the right-hand sides of formulas (19.18.6), (19.20.10), (19.20.19), (19.21.7), (19.21.8), (19.21.10), (19.25.7), (19.25.10) and (19.25.11) were written as products to ensure the correct multivalued behavior.
Reported by Luc Maisonobe on 2021-06-07