A192492 -id:A192492 - cp's OEIS frontend

A305744 Decimal expansion of imaginary part of 9th nontrivial zero of Riemann zeta function.

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

Offset: 2

Seiichi Manyama, Jun 23 2018

			The zero is at 1/2 + i * 48.0051508811671597279424727494275160416868440011444251...

Andrew M. Odlyzko, Tables of zeros of the Riemann zeta function
OEIS index entries for sequences related to the zeta function.

Imaginary part of k-th nontrivial zero of Riemann zeta function: A058303 (k=1), A065434 (k=2), A065452 (k=3), A065453 (k=4), A192492 (k=5), A305741 (k=6), A305742 (k=7), A305743 (k=8), this sequence (k=9), A306004 (k=10).

Cf. A002410 (rounded values: main entry), A013629 (floor), A092783 (ceiling).

RealDigits[Im[ZetaZero[9]], 10, 120][[1]] (* Vaclav Kotesovec, Jun 23 2018 *)

lfunzeros(1,[48,49])[1] \\ M. F. Hasler, Nov 23 2018

Edited (example, link, cross-references) by M. F. Hasler, Nov 23 2018

A306004 Decimal expansion of imaginary part of 10th nontrivial zero of Riemann zeta function.

Original entry on oeis.org

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

Offset: 2

Seiichi Manyama, Jun 23 2018

			The zero is at 1/2 + i * 49.77383247767230218191678467856372405772317829967666...

Andrew M. Odlyzko, Tables of zeros of the Riemann zeta function
OEIS index entries for sequences related to the zeta function.

Cf. A002410 (rounded values: main entry), A013629 (floor), A092783 (ceiling).

RealDigits[Im[ZetaZero[10]], 10, 120][[1]] (* Vaclav Kotesovec, Jun 23 2018 *)

lfunzeros(1,[49,50])[1] \\ M. F. Hasler, Nov 23 2018

Edited (added link, example, cross-reference) by M. F. Hasler, Nov 23 2018

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

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A305744 Decimal expansion of imaginary part of 9th nontrivial zero of Riemann zeta function.

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A306004 Decimal expansion of imaginary part of 10th nontrivial zero of Riemann zeta function.

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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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