A114856 -id:A114856 - cp's OEIS frontend

A231159 Number of Gram blocks [g(j), g(j+3)) up to 10^n with 0 <= j < 10^n.

0, 22, 779, 13822, 184107, 2169610

Offset: 3

Arkadiusz Wesolowski, Nov 04 2013

We call a Gram point g(j) "good" if j is not in A114856, and "bad" otherwise. A "Gram block of length k" is an interval [g(j), g(j+k)) such that g(j) and g(j+k) are good Gram points, g(j+1), ..., g(j+k-1) are bad Gram points, and k >= 1.

Richard P. Brent, On the Zeros of the Riemann Zeta Function in the Critical Strip, Math. Comp. 33 (1979), pp. 1361-1372.
Eric Weisstein's World of Mathematics, Gram Block
Eric Weisstein's World of Mathematics, Gram Point

Cf. A114856, A231157, A231158, A231160-A231165.

A231160 Number of Gram blocks [g(j), g(j+4)) up to 10^n with 0 <= j < 10^n.

Original entry on oeis.org

0, 19, 709, 19115, 340360

Offset: 4

Arkadiusz Wesolowski, Nov 04 2013

We call a Gram point g(j) "good" if j is not in A114856, and "bad" otherwise. A "Gram block of length k" is an interval [g(j), g(j+k)) such that g(j) and g(j+k) are good Gram points, g(j+1), ..., g(j+k-1) are bad Gram points, and k >= 1.

Richard P. Brent, On the Zeros of the Riemann Zeta Function in the Critical Strip, Math. Comp. 33 (1979), pp. 1361-1372.
Eric Weisstein's World of Mathematics, Gram Block
Eric Weisstein's World of Mathematics, Gram Point

Cf. A114856, A231157-A231159, A231161-A231165.

A231161 Number of Gram blocks [g(j), g(j+5)) up to 10^n with 0 <= j < 10^n.

Original entry on oeis.org

0, 1, 32, 821, 25813

Offset: 4

Arkadiusz Wesolowski, Nov 04 2013

We call a Gram point g(j) "good" if j is not in A114856, and "bad" otherwise. A "Gram block of length k" is an interval [g(j), g(j+k)) such that g(j) and g(j+k) are good Gram points, g(j+1), ..., g(j+k-1) are bad Gram points, and k >= 1.

Richard P. Brent, On the Zeros of the Riemann Zeta Function in the Critical Strip, Math. Comp. 33 (1979), pp. 1361-1372.
Eric Weisstein's World of Mathematics, Gram Block
Eric Weisstein's World of Mathematics, Gram Point

Cf. A114856, A231157-A231160, A231162-A231165.

A231162 Number of Gram blocks [g(j), g(j+1)) up to 10^n, 0 <= j < 10^n, which do not contain any zeros of Z(t), where Z(t) is the Riemann-Siegel Z-function.

Original entry on oeis.org

0, 42, 808, 10330, 116055, 1253556, 13197331

Offset: 2

Arkadiusz Wesolowski, Nov 04 2013

We call a Gram point g(j) "good" if j is not in A114856, and "bad" otherwise. A "Gram block of length k" is an interval [g(j), g(j+k)) such that g(j) and g(j+k) are good Gram points, g(j+1), ..., g(j+k-1) are bad Gram points, and k >= 1.

Richard P. Brent, On the Zeros of the Riemann Zeta Function in the Critical Strip, Math. Comp. 33 (1979), pp. 1361-1372.
Eric Weisstein's World of Mathematics, Gram Block
Eric Weisstein's World of Mathematics, Gram Point
Eric Weisstein's World of Mathematics, Riemann-Siegel Functions

Cf. A114856, A231157-A231161, A231163-A231165.

A231163 Number of Gram blocks [g(j), g(j+1)) up to 10^n, 0 <= j < 10^n, which contain exactly one zero of Z(t), where Z(t) is the Riemann-Siegel Z-function.

Original entry on oeis.org

1, 10, 100, 916, 8390, 79427, 769179, 7507820, 73771910

Offset: 0

Arkadiusz Wesolowski, Nov 04 2013

We call a Gram point g(j) "good" if j is not in A114856, and "bad" otherwise. A "Gram block of length k" is an interval [g(j), g(j+k)) such that g(j) and g(j+k) are good Gram points, g(j+1), ..., g(j+k-1) are bad Gram points, and k >= 1.

Richard P. Brent, On the Zeros of the Riemann Zeta Function in the Critical Strip, Math. Comp. 33 (1979), pp. 1361-1372.
Eric Weisstein's World of Mathematics, Gram Block
Eric Weisstein's World of Mathematics, Gram Point
Eric Weisstein's World of Mathematics, Riemann-Siegel Functions

Cf. A114856, A231157-A231162, A231164, A231165.

A231164 Number of Gram blocks [g(j), g(j+1)) up to 10^n, 0 <= j < 10^n, which contain exactly two zeros of Z(t), where Z(t) is the Riemann-Siegel Z-function.

Original entry on oeis.org

0, 42, 796, 10157, 113477, 1223692, 12864188

Offset: 2

Arkadiusz Wesolowski, Nov 04 2013

We call a Gram point g(j) "good" if j is not in A114856, and "bad" otherwise. A "Gram block of length k" is an interval [g(j), g(j+k)) such that g(j) and g(j+k) are good Gram points, g(j+1), ..., g(j+k-1) are bad Gram points, and k >= 1.

Richard P. Brent, On the Zeros of the Riemann Zeta Function in the Critical Strip, Math. Comp. 33 (1979), pp. 1361-1372.
Eric Weisstein's World of Mathematics, Gram Block
Eric Weisstein's World of Mathematics, Gram Point
Eric Weisstein's World of Mathematics, Riemann-Siegel Functions

Cf. A114856, A231157-A231163, A231165.

A328656 Numbers m such that d(m) < d(k) for all k < m, where d is the normalized delta defined as d(m) = (z(m+1) - z(m))(log(z(m)/(2Pi))/(2*Pi)) where z(k) is the imaginary part of the k-th Riemann zeta zero.

Original entry on oeis.org

1, 2, 4, 9, 13, 27, 34, 135, 159, 186, 212, 315, 363, 453, 693, 922, 1496, 4765, 6709, 44555, 73997, 82552, 87761, 95248, 415587, 420891, 1115578, 8546950, 24360732, 41820581, 1048449114, 3570918901, 35016977796

Offset: 1

Artur Jasinski, Jan 03 2020

nonn

No more records up to k = 103800788359.
Indices of zeros for successive maximal records of the normalized delta see A329742.
a(28)-a(33) computed by David Platt (2020).
Conjectural next terms: 1217992279429, 4088664936219.

			   n |  a(n) |    d(n)
  ---+-------+------------
   1 |     1 | 0.88871193
   2 |     2 | 0.76669277
   3 |     4 | 0.63017799
   4 |     9 | 0.57239954
   5 |    13 | 0.53062398
   6 |    27 | 0.52634271
   7 |    34 | 0.38628922
   8 |   135 | 0.37238098
   9 |   159 | 0.35780768
  10 |   186 | 0.32438582
  11 |   212 | 0.29105188
  12 |   315 | 0.24707528
  13 |   363 | 0.24343744
  14 |   453 | 0.23631515
  15 |   693 | 0.18028720
  16 |   922 | 0.13762601
  17 |  1496 | 0.08925253
  18 |  4765 | 0.04628960
  19 |  6709 | 0.04209838
  20 | 44555 | 0.04074628

Xavier Gourdon, The 1013 first zeros of the Riemann Zeta function, and zeros computation at very large height, 2004.
David Platt, Results computation of the smallest relative gaps between successive zeta zeros, 2020.

Cf. A114856, A254297, A255739, A255742, A326502, A329742.

prec = 30; min = 10; aa = {}; Do[kk = N[Im[(ZetaZero[n + 1] - ZetaZero[n])], prec] (Log[N[Im[ZetaZero[n]], prec]/(2 Pi)]/(2 Pi));
If[kk

A216700 Violations of Rosser's rule: numbers n such that the Gram block [ g(n), g(n+k) ) contains fewer than k points t such that Z(t) = 0, where Z(t) is the Riemann-Siegel Z-function.

Original entry on oeis.org

13999525, 30783329, 30930927, 37592215, 40870156, 43628107, 46082042, 46875667, 49624541, 50799238, 55221454, 56948780, 60515663, 61331766, 69784844, 75052114, 79545241, 79652248, 83088043, 83689523, 85348958, 86513820, 87947597

Offset: 1

Charles R Greathouse IV, Sep 17 2012

A Gram block [ g(m), g(m+k) ) is a half-open interval where g(m) and g(m+k) are "good" Gram points and g(m+1), ..., g(m+k-1) "bad" Gram points. A Gram point is "good" if (-1)^n Z(g(n)) > 0 and "bad" otherwise; see A114856.
Lehman showed that this sequence is infinite and conjectured (correctly) that a(1) > 10^7. Brent (1979) found a(1)-a(15). Brent, van de Lune, te Riele, & Winter extended this to a(104). Gourdon extended the calculation through a(320624341).
Trudgian showed that this sequence is of positive (lower) density.
Note: There is a typo in 7.3 of the Trudgian link showing 13999825, rather than 13999525, as the value for a(1). - Charles R Greathouse IV, Jan 27 2022

R. S. Lehman, On the distribution of zeros of the Riemann zeta-function, Proc. London Math. Soc., (3), v. 20 (1970), pp. 303-320.
J. Barkley Rosser and J. M. Yohe and Lowell Schoenfeld, Rigorous computation and the zeros of the Riemann zeta-function, Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968), vol. 1, North-Holland, Amsterdam, 1969, pp. 70-76. Errata: Math. Comp., v. 29, 1975, p. 243.

R. P. Brent, On the zeros of the Riemann zeta function in the critical strip, Math. Comp. 33 (1979), pp. 1361-1372.
R. P. Brent, J. van de Lune, H. J. J. te Riele and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip. II, Math. Comp. 39 (1982), pp. 681-688.
X. Gourdon, The 10^13 first zeros of the Riemann zeta-function and zeros computation at very large height (2004).
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.

Cf. A002505, A114856.

A327546 Indices n of j-points j(n) for successive positive maxima of the Riemann zeta function on critical line.

Original entry on oeis.org

1, 3, 6, 12, 23, 31, 39, 62, 124, 181, 211, 254, 377, 703, 869, 1207, 1443, 1702, 1933, 2565, 3968, 4657, 4803, 5815, 6618, 8569, 13879, 15321, 25461, 44681, 58716, 62728, 68865, 74399, 83452, 100050, 167369, 181802, 185011, 220569, 259499

Offset: 1

Artur Jasinski, Sep 16 2019

j-points occur when the real part of Riemann zeta function is zero but the imaginary part isn't zero.

The n-th j-point occur when Riemann-Siegel theta function is equal to Pi*(2n+1)/2.

			   n |  a(n)  | Zeta[1/2+I*j(a(n))]  |  j(a(n))
-----+--------+----------------------+------------
   1 |      1 | 0.6888099353665862*i |  25.49150821
   2 |      3 | 1.0716782759460156*i |  33.62379307
   3 |      6 | 1.3843203337013829*i |  43.99352729
   4 |     12 | 2.0558319047400831*i |  61.73354345
   5 |     23 | 2.2103659566253039*i |  89.57355850
   6 |     31 | 2.4259114706957412*i |  107.8332676
   7 |     39 | 2.5797839609135738*i |  125.0556067
   8 |     62 | 3.5676523298409918*i |  170.8597635
   9 |    124 | 3.9817183542258544*i |  279.9753243
  10 |    181 | 4.4992991376133266*i |  370.7853980
  11 |    211 | 4.7024313606767908*i |  416.3507516
  12 |    254 | 4.9763959256849833*i |  479.6816189
  13 |    377 | 6.0255895622763492*i |  651.5679685
  14 |    703 | 6.6869029304615494*i | 1068.801198
  15 |    869 | 6.9619624520146889*i | 1268.439833
  16 |   1207 | 7.0560068592571360*i | 1658.281364

Cf. A114856, A254297, A255739, A255742, A325932, A326502, A326890, A326891, A327543.

ff = 0; aa = {}; Do[kk = Im[Zeta[1/2 + I N[InverseFunction[RiemannSiegelTheta][(2 n + 1) Pi/2],10]]]; If[kk > ff, AppendTo[aa, n]; ff = kk], {n, 1, 100051}]; aa

A240961 Decimal expansion of y_126, the [imaginary part of the] first zero of Riemann's zeta function where Gram's law fails.

Original entry on oeis.org

2, 7, 9, 2, 2, 9, 2, 5, 0, 9, 2, 7, 7, 4, 5, 1, 8, 9, 2, 2, 8, 4, 0, 9, 8, 8, 0, 4, 5, 1, 9, 5, 5, 3, 5, 9, 2, 8, 3, 4, 9, 2, 6, 3, 7, 4, 0, 5, 5, 6, 1, 2, 9, 3, 5, 9, 4, 7, 2, 7, 0, 0, 9, 3, 7, 1, 3, 2, 7, 3, 1, 9, 2, 9, 1, 9, 9, 6, 3, 0, 5, 0, 7, 8, 8, 1, 6, 8, 2, 2, 5, 7, 8, 1, 9, 3, 5, 7, 9

Offset: 3

Jean-François Alcover, Aug 05 2014

			279.2292509277451892284098804519553592834926374...

H. M. Edwards, Riemann's Zeta Function, Dover Publications Inc. 1974, pp 124-127.

Eric Weisstein's MathWorld, Gram Point

Cf. A114856.

RealDigits[ZetaZero[126] // Im, 10, 99] // First

solve(y=279,279.3,real(zeta(1/2+y*I))) \\ Charles R Greathouse IV, Mar 10 2016

lfunzeros(lzeta,[279,280])[1] \\ Charles R Greathouse IV, Mar 10 2016

cp's OEIS Frontend

A231159 Number of Gram blocks [g(j), g(j+3)) up to 10^n with 0 <= j < 10^n.

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A231160 Number of Gram blocks [g(j), g(j+4)) up to 10^n with 0 <= j < 10^n.

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A231161 Number of Gram blocks [g(j), g(j+5)) up to 10^n with 0 <= j < 10^n.

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A231162 Number of Gram blocks [g(j), g(j+1)) up to 10^n, 0 <= j < 10^n, which do not contain any zeros of Z(t), where Z(t) is the Riemann-Siegel Z-function.

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A231163 Number of Gram blocks [g(j), g(j+1)) up to 10^n, 0 <= j < 10^n, which contain exactly one zero of Z(t), where Z(t) is the Riemann-Siegel Z-function.

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A231164 Number of Gram blocks [g(j), g(j+1)) up to 10^n, 0 <= j < 10^n, which contain exactly two zeros of Z(t), where Z(t) is the Riemann-Siegel Z-function.

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A328656 Numbers m such that d(m) < d(k) for all k < m, where d is the normalized delta defined as d(m) = (z(m+1) - z(m))*(log(z(m)/(2*Pi))/(2*Pi)) where z(k) is the imaginary part of the k-th Riemann zeta zero.

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A216700 Violations of Rosser's rule: numbers n such that the Gram block [ g(n), g(n+k) ) contains fewer than k points t such that Z(t) = 0, where Z(t) is the Riemann-Siegel Z-function.

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A327546 Indices n of j-points j(n) for successive positive maxima of the Riemann zeta function on critical line.

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Mathematica

A240961 Decimal expansion of y_126, the [imaginary part of the] first zero of Riemann's zeta function where Gram's law fails.

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Mathematica

A328656 Numbers m such that d(m) < d(k) for all k < m, where d is the normalized delta defined as d(m) = (z(m+1) - z(m))(log(z(m)/(2Pi))/(2*Pi)) where z(k) is the imaginary part of the k-th Riemann zeta zero.