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Dirichlet product (or convolution)

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1: 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
27.5.1 h ( n ) = d | n f ( d ) g ( n d ) ,
called the Dirichlet product (or convolution) of f and g . …
2: 25.15 Dirichlet L -functions
25.15.2 L ( s , χ ) = p ( 1 - χ ( p ) p s ) - 1 , s > 1 ,
25.15.3 L ( s , χ ) = k - s r = 1 k - 1 χ ( r ) ζ ( s , r k ) ,
25.15.4 L ( s , χ ) = L ( s , χ 0 ) p | k ( 1 - χ 0 ( p ) p s ) ,
3: 27.4 Euler Products and Dirichlet Series
§27.4 Euler Products and Dirichlet Series
4: 27.8 Dirichlet Characters
Every Dirichlet character χ (mod k ) is a product
5: 27.9 Quadratic Characters
§27.9 Quadratic Characters
The Legendre symbol ( n | p ) , as a function of n , is a Dirichlet character (mod p ). … If an odd integer P has prime factorization P = r = 1 ν ( n ) p r a r , then the Jacobi symbol ( n | P ) is defined by ( n | P ) = r = 1 ν ( n ) ( n | p r ) a r , with ( n | 1 ) = 1 . The Jacobi symbol ( n | P ) is a Dirichlet character (mod P ). …
6: 27.3 Multiplicative Properties
27.3.2 f ( n ) = r = 1 ν ( n ) f ( p r a r ) .
27.3.3 ϕ ( n ) = n p | n ( 1 - p - 1 ) ,
27.3.4 J k ( n ) = n k p | n ( 1 - p - k ) ,
27.3.5 d ( n ) = r = 1 ν ( n ) ( 1 + a r ) ,
Examples are 1 / n and λ ( n ) , and the Dirichlet characters, defined in §27.8. …
7: Bibliography D
  • C. de la Vallée Poussin (1896b) Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire M x + N . Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
  • K. Dilcher (1996) Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.
  • 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).
  • R. McD. Dodds and G. Wiechers (1972) Vector coupling coefficients as products of prime factors. Comput. Phys. Comm. 4 (2), pp. 268–274.
  • 8: 25.11 Hurwitz Zeta Function
    25.11.36 L ( s , χ ) = n = 1 χ ( n ) n s = k - s r = 1 k - 1 χ ( r ) ζ ( s , r k ) , s > 1 ,
    where χ ( n ) is a Dirichlet character ( mod k ) 27.8). …
    25.11.37 k = 1 ( - 1 ) k k ζ ( n k , a ) = - n ln Γ ( a ) + ln ( j = 0 n - 1 Γ ( a - e ( 2 j + 1 ) π i / n ) ) , n = 2 , 3 , 4 , , a 1 .
    9: Errata
  • Equation (20.4.2)
    20.4.2 θ 1 ( 0 , q ) = 2 q 1 / 4 n = 1 ( 1 - q 2 n ) 3 = 2 q 1 / 4 ( q 2 ; q 2 ) 3

    The representation in terms of ( q 2 ; q 2 ) 3 was added to this equation.

  • Equation (27.14.2)
    27.14.2 f ( x ) = m = 1 ( 1 - x m ) = ( x ; x ) , | x | < 1

    The representation in terms of ( x ; x ) was added to this equation.

  • Equation (25.11.36)

    We have emphasized the link with the Dirichlet L -function, and used the fact that χ ( k ) = 0 . A sentence just below (25.11.36) was added indicating that one should make a comparison with (25.15.1) and (25.15.3).

  • References

    Some references were added to §§7.25(ii), 7.25(iii), 7.25(vi), 8.28(ii), and to ¶Products (in §10.74(vii)) and §10.77(ix).

  • 10: 25.2 Definition and Expansions
    For further expansions of functions similar to (25.2.1) (Dirichlet series) see §27.4. …
    §25.2(iv) Infinite Products
    25.2.11 ζ ( s ) = p ( 1 - p - s ) - 1 , s > 1 ,
    product over all primes p . …product over zeros ρ of ζ with ρ > 0 (see §25.10(i)); γ is Euler’s constant (§5.2(ii)).