Search: a114857 -id:a114857
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A114856
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Indices n of ("bad") Gram points g(n) for which (-1)^n Z(g(n)) < 0, where Z(t) is the Riemann-Siegel Z-function.
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+10
24
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126, 134, 195, 211, 232, 254, 288, 367, 377, 379, 397, 400, 461, 507, 518, 529, 567, 578, 595, 618, 626, 637, 654, 668, 692, 694, 703, 715, 728, 766, 777, 793, 795, 807, 819, 848, 857, 869, 887, 964, 992, 995, 1016, 1028, 1034, 1043, 1046, 1071, 1086
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OFFSET
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1,1
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LINKS
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Eric Weisstein's World of Mathematics, Gram Point.
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FORMULA
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Trudgian shows that a(n) = O(n), that is, there exists some k such that a(n) <= k*n. - Charles R Greathouse IV, Aug 29 2012
In fact Trudgian shows that a(n) ≍ n, and further, there exist constants 1 < b < c such that b*n < a(n) < c*n. (See the paper's discussion of the Weak Gram Law.) - Charles R Greathouse IV, Mar 28 2023
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EXAMPLE
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(-1)^126 Z(g(126)) = -0.0276294988571999.... - David Baugh, Apr 02 2008
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MATHEMATICA
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g[n_] := (g0 /. FindRoot[ RiemannSiegelTheta[g0] == Pi*n, {g0, 2*Pi*Exp[1 + ProductLog[(8*n + 1)/(8*E)]]}, WorkingPrecision -> 16]); Reap[For[n = 1, n < 1100, n++, If[(-1)^n*RiemannSiegelZ[g[n]] < 0, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 17 2012, after Eric W. Weisstein *)
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PROG
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(PARI) g0(n)=2*Pi*exp(1+lambertw((8*n+1)/exp(1)/8)) \\ approximate location of gram(n)
th(t)=arg(gamma(1/4+I*t/2))-log(Pi)*t/2 \\ theta, but off by some integer multiple of 2*Pi
thapprox(t)=log(t/2/Pi)*t/2-t/2-Pi/8+1/48/t-1/5760/t^3
RStheta(t)=my(T=th(t)); (thapprox(t)-T)\/(2*Pi)*2*Pi+T
gram(n)=my(G=g0(n), k=n*Pi); solve(x=G-.003, G+1e-8, RStheta(x)-k)
Z(t)=exp(th(t)*I)*zeta(1/2+I*t)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A002505
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Nearest integer to the n-th Gram point.
(Formerly M5052 N2185)
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+10
8
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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
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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Eric Weisstein's World of Mathematics, Gram Point.
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(Sage)
a = lambda n: round(2*pi*(n - 7/8)/lambert_w((n - 7/8)/exp(1)))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A114858
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Decimal expansion of first Gram point.
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+10
3
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2, 3, 1, 7, 0, 2, 8, 2, 7, 0, 1, 2, 4, 6, 3, 0, 9, 2, 7, 8, 9, 9, 6, 6, 4, 3, 5, 3, 8, 3, 0, 1, 5, 3, 2, 0, 5, 1, 7, 4, 7, 0, 9, 8, 3, 2, 6, 8, 4, 1, 6, 4, 6, 9, 7, 0, 8, 3, 0, 0, 8, 8, 5, 1, 9, 0, 2, 2, 9, 6, 6, 0, 3, 1, 9, 9, 3, 6, 0, 9, 3, 9, 0, 3, 3, 1, 0, 5, 7, 7, 4, 8, 3, 4, 4, 6, 3, 9, 1, 6, 0, 4
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OFFSET
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2,1
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LINKS
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EXAMPLE
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23.1702827...
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MATHEMATICA
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First[ RealDigits[t /. FindRoot[ RiemannSiegelTheta[t] == Pi, {t, 23}, WorkingPrecision -> 120], 10, 102]] (* Jean-François Alcover, Jun 07 2012 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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