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The Riemann-Siegel expansion for the zeta function: high orders and remainders. (English) Zbl 0842.11030
The Riemann-Siegel expansion for the Riemann zeta function $\zeta (s)$ is a very useful representation in order to calculate with high accuracy its values on the critical line $\text{Re }s = 1/2$. The Riemann-Siegel series was deciphered by Siegel in the 1920s from Riemann’s manuscripts of the 1850s. The aim of the author is to elucidate the structure of this series in several respects. First, an alternative derivation is devised in the paper, by using formal elementary manipulations of the Dirichlet series corresponding to the zeta function, in order to obtain the terms of the expansion, which enable the author to calculate higher orders with relative ease. One should recall at this point that Riemann’s original technique for obtaining these terms (in particular, the coefficients of the remainder as written as a formal power series) is a cumbersome application of the saddle point method to an integral representation, with the subtlety that the saddle point lies on a line containing a string of poles. This contrasts very much with the formalism used by the author which, albeit formal, is very elementary. It starts by using some simple expressions obtained by {\it W. Gabcke} [`Neue Herleitung und explizite Restabschätzung der Riemann-Siegel-Formel’, Ph.D. thesis, Göttingen (1979; Zbl 0499.10040)]. The calculation of terms of higher order allows establishing the dominant behaviour of such terms, and to show in this way that the expansion is a divergent one, with the higher orders $r$ having the familiar `factorial divided by power’ dependence, affected by a slowly varying multiplier function which is calculated explicitly in the paper. The form of the remainder when the expansion is truncated near its optimal truncation term, of order $r^*$, is determined to be of order $\exp (-r^*/2)$, indicating that the critical line is a Stokes line for the Riemann-Siegel expansion. This optimal accuracy is obtained however, as confessed by the author himself, at very high computational cost. Such a procedure estimates the dependence of the truncation error on the order of the truncation, when this is large, and thence the optimal accuracy that can be obtained with the method. That is particularly interesting in view of the recent appearance of alternative methods for calculating $\zeta$ to very high accuracy [{\it M. V. Berry} and {\it J. P. Keating}, Proc. R. Soc. Lond., Ser. A 437, 151-173 (1992; Zbl 0776.11048); {\it R. B. Paris}, Proc. R. Soc. Lond., Ser. A 446, 565-587 (1994; Zbl 0827.11051)] (related results on the calculation of numerical values of Epstein zeta function with exponential accuracy can be found in the recently published monograph by the reviewer [`Ten physical applications of spectral zeta functions’ (Springer, Berlin) (1995)]. The conclusions of the paper are supported with numerical tests of the formulas, in particular, by explicit computations of the first 50 coefficients in the expansion, and of the remainders, as a function of the truncation for several values of the imginary part $t$ of the argument $s= 1/2+ it$. To conclude, it is speculated that it might be possible to attain still higher accuracy by extending `hyperasymptotics’ to the Riemann-Siegel expansion. The paper concludes with four appendices where specific aspects of the calculations are explained in detail.

11M06$\zeta (s)$ and $L(s, \chi)$
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