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The Riemann-Siegel Expansion for the Zeta Function: High Orders and Remainders

M. V. Berry
Proceedings: Mathematical and Physical Sciences
Vol. 450, No. 1939 (Aug. 8, 1995), pp. 439-462
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/52717
Page Count: 24
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The Riemann-Siegel Expansion for the Zeta Function: High Orders and Remainders
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Abstract

On the critical line s = 1/2 + it (t real), Riemann's zeta function can be calculated with high accuracy by the Riemann-Siegel expansion. This is derived here by elementary formal manipulations of the Dirichlet series. It is shown that the expansion is divergent, with the high orders r having the familiar `factorial divided by power' dependence, decorated with an unfamiliar slowly varying multiplier function which is calculated explicitly. Terms of the series decrease until r = r*≈ 2π t and then increase. The form of the remainder when the expansion is truncated near r* is determined; it is of order exp(-π t), indicating that the critical line is a Stokes line for the Riemann-Siegel expansion. These conclusions are supported by computations of the first 50 coefficients in the expansion, and of the remainders as a function of truncation for several values of t.

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