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Riemann–Siegel formula

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1: 25.10 Zeros
§25.10(ii) RiemannSiegel Formula
Sign changes of Z ( t ) are determined by multiplying (25.9.3) by exp ( i ϑ ( t ) ) to obtain the RiemannSiegel formula:
25.10.3 Z ( t ) = 2 n = 1 m cos ( ϑ ( t ) - t ln n ) n 1 / 2 + R ( t ) , m = t / ( 2 π ) ,
Calculations based on the RiemannSiegel formula reveal that the first ten billion zeros of ζ ( s ) in the critical strip are on the critical line (van de Lune et al. (1986)). …
2: 25.19 Tables
  • Cloutman (1989) tabulates Γ ( s + 1 ) F s ( x ) , where F s ( x ) is the Fermi–Dirac integral (25.12.14), for s = - 1 2 , 1 2 , 3 2 , 5 2 , x = - 5 ( .05 ) 25 , to 12S.

  • Fletcher et al. (1962, §22.1) lists many sources for earlier tables of ζ ( s ) for both real and complex s . §22.133 gives sources for numerical values of coefficients in the RiemannSiegel formula, §22.15 describes tables of values of ζ ( s , a ) , and §22.17 lists tables for some Dirichlet L -functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

  • 3: 25.18 Methods of Computation
    The principal tools for computing ζ ( s ) are the expansion (25.2.9) for general values of s , and the RiemannSiegel formula (25.10.3) (extended to higher terms) for ζ ( 1 2 + i t ) . …
    4: Bibliography B
  • A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.
  • W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.
  • M. V. Berry and J. P. Keating (1998) H = x p and the Riemann Zeros. In Supersymmetry and Trace Formulae: Chaos and Disorder, I. V. Lerner, J. P. Keating, and D. E. Khmelnitskii (Eds.), pp. 355–367.
  • M. V. Berry and J. P. Keating (1999) The Riemann zeros and eigenvalue asymptotics. SIAM Rev. 41 (2), pp. 236–266.
  • M. V. Berry (1995) The Riemann-Siegel expansion for the zeta function: High orders and remainders. Proc. Roy. Soc. London Ser. A 450, pp. 439–462.