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Computational Strategies for the Riemann Zeta Function
 Journal of Computational and Applied Mathematics
, 2000
"... We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of th ..."
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Cited by 46 (9 self)
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We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the incidence of what we call "value recycling".
The Riemann Zeros and Eigenvalue Asymptotics
 SIAM Rev
, 1999
"... Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many feat ..."
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Cited by 42 (5 self)
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Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display randommatrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The RiemannSiegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.
Fast Computation of the Riemann Zeta Function to Arbitrary Accuracy
"... s 1.5 + 1000i, N 12 0 / I and 12 13 Closeup View o 15 The Integrand Absolute Value, Real and Imaginary Parts of f(s,z) z exp(irz ) z  irz irz x + i(x 1.) 0.01 0.005 , , X 11.5 12.5 13 13.5 0.005 Quadrature Analysis a Tool Let//(tv) Note H(u .51 0.5 0 0.5 1 1 ..."
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Cited by 3 (0 self)
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s 1.5 + 1000i, N 12 0 / I and 12 13 Closeup View o 15 The Integrand Absolute Value, Real and Imaginary Parts of f(s,z) z exp(irz ) z  irz irz x + i(x 1.) 0.01 0.005 , , X 11.5 12.5 13 13.5 0.005 Quadrature Analysis a Tool Let//(tv) Note H(u .51 0.5 0 0.5 1 1.5 11 A Quadrature Formula jfc f ( z ) mCZ + f(z)H((zo + / f (z)H((z Note , similarly for f. 12 "Left Error" for /o Integrand 10.5 11 11.5 12 12.5 13 13.5 h  0.2(1 + i) 10 10.5, 12 12.5 nd 12 13 13 5 Parameters for Quadrature w N, N, N, IV, and M, where 21 Width of sample interval, and with M + I sample points NI 1/2 W(1Ii) 2W(1F i)/M 14 Quadrature Formula for /0() N M n1 m=0 n=N NT n=N+l H((N + 1/2 n)/h) n  H((n N 1/2)/h) n  where K exp(irz ) z  i?rz i?rz min(5/4 + N and No, 1/4 + N 15 Computin // to $ decimal W 2.5 M 5O Computation time using quadrature was 0.31 sec. in GP/PARI on a 300MHz Ultras
Implementing the LagariasOdlyzko Analytic Algorithm for π(x)
, 1999
"... .77> for the analytic method. so the crossover point is x = (C 2 =C 1 ) 1=(a 1 \Gammaa 2 ) Ignoring factors of x ffl we have a 1 = 2=3, a 2 = 1=2, and the crossover point is x = (C 2 =C 1 ) 6 but \Delta \Delta \Delta 2 DRAFT Sun Apr 5 19:13:05 CDT 1998 Times for MeisselLehmer : : : 15.5 ..."
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Cited by 1 (1 self)
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.77> for the analytic method. so the crossover point is x = (C 2 =C 1 ) 1=(a 1 \Gammaa 2 ) Ignoring factors of x ffl we have a 1 = 2=3, a 2 = 1=2, and the crossover point is x = (C 2 =C 1 ) 6 but \Delta \Delta \Delta 2 DRAFT Sun Apr 5 19:13:05 CDT 1998 Times for MeisselLehmer : : : 15.5 16 16.5 17 17.5 18 Log [x] 10 3.75 4.25 4.5 4.75 5 5.25 5.5 Log [t] 10 LogLog Fit to DelegliseRivat timings log 10 (t) = \Gamma5:768 + 0:626 log 10 (x) (t in seconds
DYNAMICAL ZETA FUNCTIONS AND KUMMER CONGRUENCES
, 2003
"... Abstract. We establish a connection between the coefficients of ArtinMazur ..."
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Cited by 1 (0 self)
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Abstract. We establish a connection between the coefficients of ArtinMazur
Computational Number Theory at CWI in 19701994
, 1994
"... this paper we present a concise survey of the research in Computational ..."