Gram Point

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