Review of Crary and Rosser

F. D. Crary and J. Barkley Rosser, High precision coefficients related to the zeta function, reviewed in Mathematics of Computation 31 (1977), 803-804. [pdf]

Comments (December 2006)

The representation of the coefficients in the Riemann-Siegel formula given in my review was suggested by Lehmer (reference [4] of the review).

The sequence (rho_n) = (1, 2, 82, 10572, ...) satisfies a recurrence in terms of Euler numbers and is Sloane's sequence A087617. Sloane calls it the Gabcke sequence but Gabcke credits Lehmer (Gabcke's reference [8]) so perhaps it should be called the Lehmer-Gabcke sequence.

The sequence (rho_n) occurs naturally in the asymptotic expansion of ln(Gamma(1/4 + i.t/2)). It is not obvious from the recurrence that the rho_n are integers, but this was proved by Juan Arias de Reyna.

The Riemann-Siegel formula is an asymptotic expansion with inherent error of order exp(-pi.t). I observed this empirically at the time of writing the review. Berry (1995) gives a plausible argument (not rigorous since it uses divergent series). In practice it is usually sufficient to take a small number of terms in the remainder R(t) in combination with Gabcke's rigorous error bounds.

Thanks to David Broadhurst for his comments and Juan Arias de Reyna for his translation of Gabcke.

References

M. V. Berry, The Riemann-Siegel expansion for the zeta function: high orders and remainders, Proc. R. Soc. Lond. A (1995) 450, 439-462.

W. Gabcke, Neue Herleitung und explizite Restäbschatzung der Riemann-Siegel-Formel, Dissertation, Univ. Göttingen, 1979.

D. H. Lehmer, Extended computation of the Riemann zeta-function, Mathematika 3 (1956), 102-108.

J. Arias de Reyna, Dynamical zeta functions and Kummer congruences, ArXiv math.NT/0309190, 2003.

J. Arias de Reyna, High precision computation of Riemann's zeta function by the Riemann-Siegel Formula, I, Mathematics of Computation 80 (2011), 995-1009. [Note: this paper is not restricted to computation on the critical line.]

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