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Riemann ξ-function

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1: 25.20 Approximations
  • Antia (1993) gives minimax rational approximations for Γ ( s + 1 ) F s ( x ) , where F s ( x ) is the Fermi–Dirac integral (25.12.14), for the intervals - < x 2 and 2 x < , with s = - 1 2 , 1 2 , 3 2 , 5 2 . For each s there are three sets of approximations, with relative maximum errors 10 - 4 , 10 - 8 , 10 - 12 .

  • 2: 25.3 Graphics
    See accompanying text
    Figure 25.3.1: Riemann zeta function ζ ( x ) and its derivative ζ ( x ) , - 20 x 10 . Magnify
    See accompanying text
    Figure 25.3.2: Riemann zeta function ζ ( x ) and its derivative ζ ( x ) , - 12 x - 2 . Magnify
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    Figure 25.3.3: Modulus of the Riemann zeta function | ζ ( x + i y ) | , - 4 x 4 , - 10 y 40 . Magnify 3D Help
    3: 25.4 Reflection Formulas
    25.4.3 ξ ( s ) = ξ ( 1 - s ) ,
    4: 25.10 Zeros
    The functional equation (25.4.1) implies ζ ( - 2 n ) = 0 for n = 1 , 2 , 3 , . …
    5: 8.22 Mathematical Applications
    8.22.2 ζ x ( s ) = 1 Γ ( s ) 0 x t s - 1 e t - 1 d t , s > 1 ,
    6: 27.4 Euler Products and Dirichlet Series
    The completely multiplicative function f ( n ) = n - s gives the Euler product representation of the Riemann zeta function ζ ( s ) 25.2(i)): …
    27.4.10 n = 1 d k ( n ) n - s = ( ζ ( s ) ) k , s > 1 ,
    7: 5.15 Polygamma Functions
    §5.15 Polygamma Functions
    The functions ψ ( n ) ( z ) , n = 1 , 2 , , are called the polygamma functions. …Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … In (5.15.2)–(5.15.7) n , m = 1 , 2 , 3 , , and for ζ ( n + 1 ) see §25.6(i). … For B 2 k see §24.2(i). …
    8: 26.12 Plane Partitions
    The complement of π B ( r , s , t ) is π c = { ( h , j , k ) | ( r - h + 1 , s - j + 1 , t - k + 1 ) π } . …
    §26.12(ii) Generating Functions
    26.12.26 pp ( n ) ( ζ ( 3 ) ) 7 / 36 2 11 / 36 ( 3 π ) 1 / 2 n 25 / 36 exp ( 3 ( ζ ( 3 ) ) 1 / 3 ( 1 2 n ) 2 / 3 + ζ ( - 1 ) ) ,
    where ζ is the Riemann ζ -function25.2(i)). …
    ζ ( - 1 ) = - 0.16542 11437 .
    9: 21.4 Graphics
    §21.4 Graphics
    Figure 21.4.1 provides surfaces of the scaled Riemann theta function θ ^ ( z | Ω ) , with …This Riemann matrix originates from the Riemann surface represented by the algebraic curve μ 3 - λ 7 + 2 λ 3 μ = 0 ; compare §21.7(i). … For the scaled Riemann theta functions depicted in Figures 21.4.221.4.5
    See accompanying text
    Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: θ ^ ( x + i y , 0 , 0 | Ω 2 ) , 0 x 1 , 0 y 3 . … Magnify 3D Help