A161914 -id:A161914 - cp's OEIS frontend

A221974 Numbers n such that A161914(n) = 0.

187, 213, 299, 316, 364, 379, 437, 454, 478, 486, 509, 580, 607, 620, 644, 670, 694, 696, 717, 752, 795, 846, 850, 871, 884, 890, 906, 923, 937, 939, 944, 966, 986, 997, 1030, 1045, 1048, 1096, 1135, 1150, 1158, 1167, 1181, 1209, 1229, 1233, 1239, 1252, 1272

Offset: 1

Omar E. Pol, Feb 02 2013

nonn

Sequence related to nontrivial zeros of Riemann zeta function.

Cf. A002410, A161914, A208436.

Position[Table[Round@Im[ZetaZero[n] - ZetaZero[n - 1]], {n, 2, 509}], 0] + 1 // Flatten (* Arkadiusz Wesolowski, Feb 05 2013 *)

a(n) ~ n. There are constants N and k such that for all n > N, a(n) = n + k. - Charles R Greathouse IV, Apr 18 2013

a(35)-a(49) from Arkadiusz Wesolowski, Feb 05 2013

A254297 Consider the nontrivial zeros of the Riemann zeta function on the critical line 1/2 + i*t and the gap, or first difference, between two consecutive such zeros; a(n) is the lesser of the two zeros at a place where the gap attains a new minimum.

Original entry on oeis.org

1, 2, 3, 5, 8, 10, 14, 20, 25, 28, 35, 64, 72, 92, 136, 160, 187, 213, 299, 316, 364, 454, 694, 923, 1497, 3778, 4766, 6710, 18860, 44556, 73998, 82553, 87762, 95249, 354770, 415588, 420892, 1115579, 8546951

Offset: 1

Robert G. Wilson v, Jan 27 2015

nonn

Since all zeros are assumed to be on the critical line, the gap, or first difference, between two consecutive zeros is measured as the difference between the two imaginary parts.
Inspired by A002410.
No other terms < 10000000. The minimum gap so far is 0.002323...

			a(1)=1 since the first Riemann zeta zero, 1/2 + i*14.13472514... (A058303) has no previous zero, so its gap is measured from 0.
a(2)=2 since the second Riemann zeta zero, 1/2 + i*21.02203964... (A065434) has a gap of 6.887314497... which is less than the previous gap of ~14.13472514.
a(3)=3 since the third Riemann zeta zero, 1/2 + i*25.01085758... (A065452) has a gap of 3.988817941... which is less than ~6.887314497.
The fourth Riemann zeta zero, 1/2 + i*30.42487613... (A065453) has a gap of 5.414018546... which is not less than ~6.887314497 and therefore is not in the sequence.
a(4)=5 since the fifth Riemann zeta zero, 1/2 + i*32.93506159... (A192492) has a gap of 2.510185462... which is less than ~3.988817941.
a(5)=8 since the eighth Riemann zeta zero, 1/2 + i*43.32707328...  has a gap of 2.408354269... which is less than ~2.510185462.

Glen Pugh, The Riemann Hypothesis in a Nutshell.

Cf. A002410, A100060, A161914, A117538, A153595, A326502.

k = 1; mn = Infinity; y = 0; lst = {}; While[k < 10001, z = N[ Im@ ZetaZero@ k, 64]; If[z - y < mn, mn = z - y; AppendTo[lst, k]]; y = z; k++]; lst

a(n) = A326502(n) + 1. - Artur Jasinski, Oct 24 2019

a(38) from Arkadiusz Wesolowski, Nov 08 2015

a(39) from Artur Jasinski, Oct 24 2019

A208436 Indices of maximal gaps between consecutive nontrivial zeros of the Riemann zeta function.

Original entry on oeis.org

1, 3, 8, 14, 33, 64, 79, 126, 183, 379, 795, 1935, 2292, 3296, 4805, 6620, 15323, 19187, 20105, 36719, 46589, 185013, 220571, 259501, 516200, 880694, 1493008, 1663325, 1793281, 3206674, 6488753, 14145077, 22653912, 33742399, 65336924, 70354407, 81805537, 110280572, 129842508, 298466597, 566415148

Offset: 1

Charles R Greathouse IV, Feb 26 2012

Values are conjectural: in principle exact values can be computed, but the bound involves a triple logarithm (or double logarithm under RH) and so is computationally infeasible.
Corresponding gap sizes are (to 7 decimal places) 6.887314, 5.414019, 4.678078, 4.280766, 3.860924, 3.499560, 3.249442, 3.235864, 3.011009, 2.980888, 2.594919, 2.589789, 2.539380, 2.406590, 2.279428, 2.194404, 2.176083, 2.098284, 2.064198, 2.042407, 2.024333, 1.966653, 1.844023, 1.804885, 1.798398, 1.779155, 1.754010, 1.696635, 1.688765, 1.686034, 1.580157, 1.567382, 1.525555, 1.521410, 1.488847, 1.479976, 1.432771, 1.422617, 1.420599, 1.413245, 1.393242, ....
Goldston & Gonek show, on the Riemann Hypothesis, that the gap between the zero 1/2 + ix and the following zero is at most Pi(1 + o(1))/log log x. For illustrative purposes, if o(1) is taken to be zero, it would suffice to check up to height 345.9 to verify a(24) with gap size 1.804... but over 10^10 for gap size 1. - Charles R Greathouse IV, Oct 22 2012
Littlewood proves an unconditional version: there is some constant Y such that the gap between the zero x + i*y and the following zero is at most 32/log log log y for all y > Y. Hall & Hayman improve the above constant from 32 to Pi/2 + o(1). Even with this latter improvement, verifying a gap of size 1 with this formula (even dropping the o(1)) would take a height above 10^53. - Charles R Greathouse IV, Jun 04 2021
Ivić improves the Goldston & Gonek constant to Pi/2. - Charles R Greathouse IV, Jun 23 2021
Bui & Milinovich, improving on Bredberg, prove that, for large enough T, there is a gap of length 6.36*Pi/log T between T and 2T (that is, 3.18 times the length of the average gap). This seems rather sharp around values where we're computing this sequence. - Charles R Greathouse IV, Jun 24 2021
Simonič (2018) shows that a(1) = 1, see the proof of Lemma 3 around (8). - Charles R Greathouse IV, Jul 07 2021
Simonič (2022) shows that, under the Riemann hypothesis, it suffices to check up to height 10^2465 to prove that the values of a(2)-a(40) are as stated. a(41) requires checking to 10^2588. Because these values are so high it is not currently feasible to prove more terms of the sequence, even under RH. - Charles R Greathouse IV, Mar 28 2022

			The first four nontrivial zeros of the zeta function are at 0.5 + 14.13472...i, 0.5 + 21.02203...i, 0.5 + 25.01085...i, and 0.5 + 30.42487...i with gaps 6.88731..., 3.98882..., and 5.41402....  No gap is larger than the first, so a(1) = 1.  The third gap is larger than all gaps later than the first gap, so a(2) = 3.

R. R. Hall and W. K. Hayman, Hyperbolic distance and distinct zeros of the Riemann zeta-function in small regions, Journal für die reine und angewandte Mathematik vol. 526 (2000), pp. 35-59.
J. E. Littlewood, Two notes on the Riemann zeta-function, Proceedings of the Cambridge Philosophical Society Vol. 22, No. 3 (1924), pp. 234-242.
E. C. Titchmarsh, The theory of the Riemann zeta-function, 1986.

H. M. Bui and M. B. Milinovich, Gaps between zeros of the Riemann zeta-function, 2017 preprint. arXiv:1410.3635 [math.NT]
Johan Bredberg, Large gaps between consecutive zeros, on the critical line, of the Riemann zeta-function, 2011 preprint. arXiv:1101.3197 [math.NT]
D. A. Goldston and S. M. Gonek, A note on S(t) and the zeros of the Riemann zeta-function, Bull. London Math. Soc. 39 (2007), pp. 482-486.
Aleksandar Ivić, Some remarks on the differences between ordinates of consecutive zeta zeros, Funct. Approx. Comment. Math. 58:1 (2018), pp. 23-35. arXiv:1610.01317 [math.NT]
LMFDB, Zeros of ζ(s)
Andrew Odlyzko, Tables of zeros of the Riemann zeta function
Aleksander Simonič, Lehmer pairs and derivatives of Hardy's Z-function, Journal of Number Theory, Vol. 184 (March 2018), pp. 451-460. arXiv:1612.08627 [math.NT]
Aleksander Simonič, On explicit estimates for S(t), S1(t), and ζ(1/2 + it) under the Riemann Hypothesis, Journal of Number Theory, Vol. 231 (February 2022), pp. 464-491. arXiv:2010.13307 [math.NT]
Index entries for zeta function

Cf. A161914, A002410.

a(27)-a(41) from Andrey V. Kulsha, Aug 27 2012

A162774 14 together with the first differences of A002410.

Original entry on oeis.org

14, 7, 4, 5, 3, 5, 3, 2, 5, 2, 3, 3, 3, 2, 4, 2, 3, 2, 4, 1, 2, 4, 2, 2, 2, 3, 3, 1, 3, 2, 3, 1, 2, 4, 1, 2, 2, 3, 2, 2, 1, 4, 2, 1, 2, 2, 3, 2, 1, 2, 3, 1, 3, 1, 2, 3, 2, 1, 2, 2, 3, 1, 2, 1, 3, 2, 1, 2, 2, 2, 3, 1, 1, 2, 3, 1, 2, 2, 1, 3, 1, 2, 1, 3, 2, 2, 1, 2, 1, 3, 2, 0, 3, 1, 2, 2, 2, 1, 2, 3, 1, 2, 1, 2, 1, 3

Offset: 1

Omar E. Pol, Jul 13 2009

nonn

This sequence is related to the Riemann zeta function (very similar to A161914).

All but finitely many terms of this sequence are 0. What is the largest n such that a(n) > 0? - Charles R Greathouse IV, Jul 16 2012

Index entries for zeta function

Cf. A002410, A161914, A162780, A162781, A162782.

More terms (data table at A161914) from Hagen von Eitzen, Oct 03 2009
a(92)-a(105) corrected by Omar E. Pol, Oct 08 2009
Missing a(92)=0 inserted by Sean A. Irvine, Mar 03 2023

A162781 a(n) = A002410(n) - A162780(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1

Offset: 1

Omar E. Pol, Jul 13 2009

sign

This sequence is related to the Riemann zeta function.

Index entries for zeta function

Cf. A002410, A162774, A161914, A162780, A162782.

More terms from Jinyuan Wang, Mar 02 2020

A210447 Number of primes <= Im(rho_n), where rho_n is the n-th nontrivial zero of Riemann zeta function.

Original entry on oeis.org

6, 8, 9, 10, 11, 12, 12, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 26, 27, 27, 28, 29, 29, 30, 30, 30, 30, 30, 30, 31, 31, 32, 32, 32, 33, 34, 34, 34, 34, 34, 35, 35, 36, 36, 37, 37, 37, 38, 38, 39, 39, 39, 40, 40

Offset: 1

Omar E. Pol, Feb 03 2013

nonn

The zeros -2, -4, -6, ... of the Riemann zeta function are considered trivial. The nontrivial zeros are in the "critical strip" 0 < Re(rho_n) < 1. All of the known nontrivial zeros have real part 1/2. In this sequence, we count the prime numbers less than or equal to the imaginary part of these nontrivial zeros.

The Riemann hypothesis (currently unproven) states that all of the nontrivial zeros have real part 1/2.

			a(8) = 12 because the 8th nontrivial zero of Riemann zeta function is 0.5 + (40.91...)i and there are 12 primes less than or equal to 40.91...; they are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and 37.

A. Odlyzko, Tables of zeros of the Riemann zeta function
Index entries for zeta function

Cf. A000720, A002410, A161914, A208436.

f[n_] := PrimePi@ Im@ ZetaZero@ n; Array[f, 70] (* Robert G. Wilson v, Jan 27 2015 *)

cp's OEIS Frontend

A162780 Partial sums of A161914.

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A162782 a(n) = A162774(n) - A161914(n).

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A221974 Numbers n such that A161914(n) = 0.

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A254297 Consider the nontrivial zeros of the Riemann zeta function on the critical line 1/2 + i*t and the gap, or first difference, between two consecutive such zeros; a(n) is the lesser of the two zeros at a place where the gap attains a new minimum.

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Mathematica

Formula

Extensions

A208436 Indices of maximal gaps between consecutive nontrivial zeros of the Riemann zeta function.

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A162774 14 together with the first differences of A002410.

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A162781 a(n) = A002410(n) - A162780(n).

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A210447 Number of primes <= Im(rho_n), where rho_n is the n-th nontrivial zero of Riemann zeta function.

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Mathematica