A002505 Nearest integer to the n-th Gram point.

18, 23, 28, 32, 35, 39, 42, 46, 49, 52, 55, 58, 60, 63, 66, 68, 71, 74, 76, 79, 81, 84, 86, 88, 91, 93, 95, 98, 100, 102, 104, 107, 109, 111, 113, 115, 118, 120, 122, 124, 126

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Every integer greater than 3295 is in this sequence. - T. D. Noe, Aug 03 2007

Nearest integer to points t such that Re(zeta(1/2+i*t)) is not equal to zero and Im(zeta(1/2+i*t))=0. - Mats Granvik, May 14 2016

References

• C. B. Haselgrove and J. C. P. Miller, Tables of the Riemann Zeta Function. Royal Society Mathematical Tables, Vol. 6, Cambridge Univ. Press, 1960, p. 58.
• A. Ivić, The Theory of Hardy's Z-Function, Cambridge University Press, 2013, pages 109-112.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..3000
• Guilherme França and André LeClair, A theory for the zeros of Riemann Zeta and other L-functions, arXiv:1407.4358 [math.NT], 2014, formula (163) at page 47.
• Mats Granvik Gram points computed by iterative formula.
• C. B. Haselgrove and J. C. P. Miller, Tables of the Riemann Zeta Function, annotated scanned copy of page 58.
• Eric Weisstein's World of Mathematics, Gram Point.

Crossrefs

Cf. A273061. A114857 = 17.8455995..., A114858 = 23.1702827...

Programs

Mathematica

a[n_] := Round[ g /. FindRoot[ RiemannSiegelTheta[g] == Pi*n, {g, 2*Pi*Exp[1 + ProductLog[(8*n + 1)/(8*E)]]}]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 17 2012, after Eric W. Weisstein *)

Sage

a = lambda n: round(2*pi*(n - 7/8)/lambert_w((n - 7/8)/exp(1)))
print([a(n) for n in (1..41)]) # Peter Luschny, May 19 2016

Formula

a(n) ~ 2*Pi*n/log n. - Charles R Greathouse IV, Oct 23 2015

From Mats Granvik, May 16 2016: (Start)

a(n) = round(2*Pi*exp(1 + LambertW((8*n + 1)/(8*exp(1))))), Eric Weisstein's World of Mathematics.

a(n+1) = round(2*Pi*(n - 7/8)/LambertW((n - 7/8)/exp(1))), after Guilherme França, André LeClair formula (163) page 47.

(End)

For c = 0 the n-th Gram point x is the fixed point solution to the iterative formula:

x = 2*Pi*e^(LambertW (-((c - n + RiemannSiegelTheta (x)/Pi + (x*(-log (x) + 1 + log (2) + log (Pi)))/(2*Pi) + 2)/e)) + 1). - Mats Granvik, Jun 17 2017