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A002505 Nearest integer to the n-th Gram point.
(Formerly M5052 N2185)
8

%I M5052 N2185

%S 18,23,28,32,35,39,42,46,49,52,55,58,60,63,66,68,71,74,76,79,81,84,86,

%T 88,91,93,95,98,100,102,104,107,109,111,113,115,118,120,122,124,126

%N Nearest integer to the n-th Gram point.

%C Every integer greater than 3295 is in this sequence. - _T. D. Noe_, Aug 03 2007

%C 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

%D 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.

%D A. Ivić, The Theory of Hardy's Z-Function, Cambridge University Press, 2013, pages 109-112.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002505/b002505.txt">Table of n, a(n) for n = 0..3000</a>

%H Guilherme França and André LeClair, <a href="http://arxiv.org/abs/1407.4358">A theory for the zeros of Riemann Zeta and other L-functions</a>, arXiv:1407.4358 [math.NT], 2014, formula (163) at page 47.

%H Mats Granvik <a href="https://pastebin.com/cZ0Ue7rN">Gram points computed by iterative formula</a>

%H C. B. Haselgrove and J. C. P. Miller, <a href="/A002505/a002505.pdf">Tables of the Riemann Zeta Function</a>, annotated scanned copy of page 58.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GramPoint.html">Gram Point</a>

%F a(n) ~ 2*Pi*n/log n. - _Charles R Greathouse IV_, Oct 23 2015

%F From _Mats Granvik_, May 16 2016: (Start)

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

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

%F (End)

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

%F 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

%t 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_ *)

%o (Sage)

%o a = lambda n: round(2*pi*(n - 7/8)/lambert_w((n - 7/8)/exp(1)))

%o print([a(n) for n in (1..41)]) # _Peter Luschny_, May 19 2016

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

%K nonn

%O 0,1

%A _N. J. A. Sloane_.

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Last modified July 29 09:41 EDT 2021. Contains 346344 sequences. (Running on oeis4.)