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1: 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 ) . …
2: 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 …
27.5.6 G ( x ) = n x F ( x n ) F ( x ) = n x μ ( n ) G ( x n ) ,
27.5.7 G ( x ) = m = 1 F ( m x ) m s F ( x ) = m = 1 μ ( m ) G ( m x ) m s ,
3: 25.21 Software
§25.21(ix) Dirichlet L -series
4: 25.2 Definition and Expansions
For further expansions of functions similar to (25.2.1) (Dirichlet series) see §27.4. …
5: 25.15 Dirichlet L -functions
The notation L ( s , χ ) was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series
25.15.1 L ( s , χ ) = n = 1 χ ( n ) n s , s > 1 ,
6: Bibliography
  • T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.
  • T. M. Apostol (1990) Modular Functions and Dirichlet Series in Number Theory. 2nd edition, Graduate Texts in Mathematics, Vol. 41, Springer-Verlag, New York.
  • 7: Bibliography M
  • H. Maass (1971) Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.
  • 8: Software Index
    9: 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 . … is a periodic function of n ( mod k ) and has the finite Fourier-series expansion … 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 … The finite Fourier expansion of a primitive Dirichlet character χ ( mod k ) has the form …
    10: Bibliography D
  • P. G. L. Dirichlet (1837) 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).
  • P. G. L. Dirichlet (1849) Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1849, pp. 69–83 (German).