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Date: 9-10-2019
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Date: 25-8-2018
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Date: 22-8-2019
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Let be the Riemann-Siegel function. The unique value such that
(1) |
where , 1, ... is then known as a Gram point (Edwards 2001, pp. 125-126).
An excellent approximation for Gram point can be obtained by using the first few terms in the asymptotic expansion for and inverting to obtain
(2) |
where is the Lambert W-function. This approximation gives as error of for , decreasing to by .
The following table gives the first few Gram points.
OEIS | ||
0 | A114857 | 17.8455995404 |
1 | A114858 | 23.1702827012 |
2 | 27.6701822178 | |
3 | 31.7179799547 | |
4 | 35.4671842971 | |
5 | 38.9992099640 | |
6 | 42.3635503920 | |
7 | 45.5930289815 | |
8 | 48.7107766217 | |
9 | 51.7338428133 | |
10 | 54.6752374468 |
The integers closest to these points are 18, 23, 28, 32, 35, 39, 42, 46, 49, 52, 55, 58, ... (OEIS A002505).
There is a unique point at which , given by the solution to the equation
(3) |
and having numerical value
(4) |
(OEIS A114893).
It is usually the case that . Values of for which this does not hold are , 134, 195, 211, 232, 254, 288, ... (OEIS A114856), the first two of which were found by Hutchinson (1925).
REFERENCES:
Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.
Gram, J.-P. "Sur les zéros de la fonction de Riemann." Acta Math. 27, 289-304, 1903.
Haselgrove, C. B. and Miller, J. C. P. "Tables of the Riemann Zeta Function." Royal Society Mathematical Tables, Vol. 6. Cambridge, England: Cambridge University Press, p. 58, 1960.
Hutchinson, J. I. "On the Roots of the Riemann Zeta-Function." Trans. Amer. Math. Soc. 27, 49-60, 1925.
Sloane, N. J. A. Sequences A002505/M5052, A114856, A114857, A114858, and A114893 Sloane, N. J. A. Sequences
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