About the Project
NIST

Riemann zeta function

AdvancedHelp

(0.006 seconds)

1—10 of 58 matching pages

1: 25.1 Special Notation
k , m , n nonnegative integers.
The main function treated in this chapter is the Riemann zeta function ζ ( s ) . … The main related functions are the Hurwitz zeta function ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .
2: 8.22 Mathematical Applications
§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function
If ζ x ( s ) denotes the incomplete Riemann zeta function defined by …so that lim x ζ x ( s ) = ζ ( s ) , then
8.22.3 ζ x ( s ) = k = 1 k - s P ( s , k x ) , s > 1 .
For further information on ζ x ( s ) , including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). …
3: 25.7 Integrals
§25.7 Integrals
For definite integrals of the Riemann zeta function see Prudnikov et al. (1986b, §2.4), Prudnikov et al. (1992a, §3.2), and Prudnikov et al. (1992b, §3.2).
4: 25.17 Physical Applications
§25.17 Physical Applications
See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). …
5: 25.20 Approximations
  • Cody et al. (1971) gives rational approximations for ζ ( s ) in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are 0.5 s 5 , 5 s 11 , 11 s 25 , 25 s 55 . Precision is varied, with a maximum of 20S.

  • Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of s ζ ( s + 1 ) and ζ ( s + k ) , k = 2 , 3 , 4 , 5 , 8 , for 0 s 1 (23D).

  • 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 .

  • 6: 25.3 Graphics
    §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
    See accompanying text
    Figure 25.3.3: Modulus of the Riemann zeta function | ζ ( x + i y ) | , - 4 x 4 , - 10 y 40 . Magnify 3D Help
    See accompanying text
    Figure 25.3.6: Z ( t ) , 10000 t 10050 . Magnify
    7: 25.18 Methods of Computation
    §25.18(i) Function Values and Derivatives
    Calculations relating to derivatives of ζ ( s ) and/or ζ ( s , a ) can be found in Apostol (1985a), Choudhury (1995), Miller and Adamchik (1998), and Yeremin et al. (1988). …
    §25.18(ii) Zeros
    Most numerical calculations of the Riemann zeta function are concerned with locating zeros of ζ ( 1 2 + i t ) in an effort to prove or disprove the Riemann hypothesis, which states that all nontrivial zeros of ζ ( s ) lie on the critical line s = 1 2 . …
    8: 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)): … The Riemann zeta function is the prototype of series of the form …
    27.4.5 n = 1 μ ( n ) n - s = 1 ζ ( s ) , s > 1 ,
    27.4.7 n = 1 λ ( n ) n - s = ζ ( 2 s ) ζ ( s ) , s > 1 ,
    27.4.12 n = 1 Λ ( n ) n - s = - ζ ( s ) ζ ( s ) , s > 1 ,
    9: 25.4 Reflection Formulas
    §25.4 Reflection Formulas
    25.4.1 ζ ( 1 - s ) = 2 ( 2 π ) - s cos ( 1 2 π s ) Γ ( s ) ζ ( s ) ,
    25.4.2 ζ ( s ) = 2 ( 2 π ) s - 1 sin ( 1 2 π s ) Γ ( 1 - s ) ζ ( 1 - s ) .
    25.4.4 ξ ( s ) = 1 2 s ( s - 1 ) Γ ( 1 2 s ) π - s / 2 ζ ( s ) .
    25.4.5 ( - 1 ) k ζ ( k ) ( 1 - s ) = 2 ( 2 π ) s m = 0 k r = 0 m ( k m ) ( m r ) ( ( c k - m ) cos ( 1 2 π s ) + ( c k - m ) sin ( 1 2 π s ) ) Γ ( r ) ( s ) ζ ( m - r ) ( s ) ,
    10: 25.8 Sums
    §25.8 Sums
    25.8.1 k = 2 ( ζ ( k ) - 1 ) = 1 .
    25.8.2 k = 0 Γ ( s + k ) ( k + 1 ) ! ( ζ ( s + k ) - 1 ) = Γ ( s - 1 ) , s 1 , 0 , - 1 , - 2 , .
    25.8.4 k = 1 ( - 1 ) k k ( ζ ( n k ) - 1 ) = ln ( j = 0 n - 1 Γ ( 2 - e ( 2 j + 1 ) π i / n ) ) , n = 2 , 3 , 4 , .
    25.8.9 k = 1 ζ ( 2 k ) ( 2 k + 1 ) 2 2 k = 1 2 - 1 2 ln 2 .