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URL: https://oeis.org/A002505

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A002505
Nearest integer to the n-th Gram point.
8
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, 128, 130, 132, 134, 136, 138, 141, 143, 145, 147, 149, 151, 152, 154, 156, 158, 160
OFFSET
0,1
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
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.
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.
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
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 *)
PROG
(SageMath)
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
CROSSREFS
Cf. A273061. A114857 = 17.8455995..., A114858 = 23.1702827...
Sequence in context: A339473 A361627 A391020 * A182438 A050772 A376507
KEYWORD
nonn
STATUS
approved