Abstract
On the critical line
s
═ ½ + i
t
(
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
.
Reference19 articles.
1. n t )found in §4.
2. Berry M. V. 1986 In Quantum chaos and statistical nuclear physics (ed. T. H. Seligman &; H. Nishioka) pp. 1-17. Springer Lecture Notes in Physics No. 263.
3. Uniform asymptotic smoothing of Stokes’s discontinuities
4. Infinitely many Stokes smoothings in the gamma function
5. Hyperasymptotics for integrals with saddles
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