A114857 -id:A114857 - cp's OEIS frontend

A114856 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.

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

Offset: 1

Eric W. Weisstein, Jan 02 2006

nonn

			(-1)^126 Z(g(126)) = -0.0276294988571999.... - _David Baugh_, Apr 02 2008

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Mathematics Stack Exchange, crow, Can we prove B(n)=(1/4)G(n-1)G(n) is an indicator function ...
E. C. Titchmarsh, On van der Corput's Method and the zeta-function of Riemann IV, Quarterly Journal of Mathematics os-5 (1934), pp. 98-105.
Timothy Trudgian, On the success and failure of Gram's Law and the Rosser Rule, Acta Arithmetica, 2011 | 148 | 3 | 225-256.
Eric Weisstein's World of Mathematics, Gram Point.

Cf. A114857, A114858, A216700.

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 *)

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)
is(n)=my(G=gram(n));real((-1)^n*Z(G))<0 \\ Charles R Greathouse IV, Jan 22 2022

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

Definition corrected by David Baugh, Apr 02 2008

A002505 Nearest integer to the n-th Gram point.

Original entry on oeis.org

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

N. J. A. Sloane

nonn

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

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).

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.

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

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 *)

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

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

A114858 Decimal expansion of first Gram point.

Original entry on oeis.org

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

Offset: 2

Eric W. Weisstein, Jan 02 2006

			23.1702827...

Eric Weisstein's World of Mathematics, Gram Point

Cf. A114856, A114857.

First[ RealDigits[t /. FindRoot[ RiemannSiegelTheta[t] == Pi, {t, 23}, WorkingPrecision -> 120], 10, 102]] (* Jean-François Alcover, Jun 07 2012 *)

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A114856 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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A002505 Nearest integer to the n-th Gram point.

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A114858 Decimal expansion of first Gram point.

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