A065453 -id:A065453 - 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.

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), A305744 (k=9), this sequence (k=10).

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

A131586 Concatenation of first n numbers of the decimal expansion of imaginary part of 4th nontrivial zero of Riemann zeta function.

Original entry on oeis.org

3, 30, 304, 3042, 30424, 304248, 3042487, 30424876, 304248761, 3042487612, 30424876125, 304248761258, 3042487612585, 30424876125859, 304248761258595, 3042487612585951, 30424876125859513, 304248761258595132

Offset: 0

Omar E. Pol, Sep 13 2007

Index entries for zeta function.

See A002410 and A065453 for more information.

Module[{nn=30,zzero},zzero=RealDigits[Im[N[ZetaZero[4],nn]]][[1]];Table[ FromDigits[Take[zzero,n]],{n,nn}]] (* Harvey P. Dale, Jul 06 2021 *)

A133010 Characteristic function of the Riemann zeta function: If n is a nearest integer to imaginary part of zero, then a(n)=1 else a(n)=0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0

Offset: 1

Omar E. Pol, Sep 13 2007

Also an interesting triangle read by row: See tabl.

			a(30)=1 because 30 is the nearest integer to imaginary part of 4th nontrivial zero of Riemann zeta function.

See A002410 for more information. Cf. A058303, A065434, A065452, A065453.

A133011 Characteristic function of the Riemann zeta function: If n is a nearest integer to imaginary part of zero, then a(n)=0 else a(n)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1

Offset: 1

Omar E. Pol, Sep 13 2007

Also an interesting triangle read by row: See tabl.

			a(30)=0 because 30 is the nearest integer to imaginary part of 4th nontrivial zero of Riemann zeta function.

See A002410 for more information. Cf. A058303, A065434, A065452, A065453.

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A131586 Concatenation of first n numbers of the decimal expansion of imaginary part of 4th nontrivial zero of Riemann zeta function.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A133010 Characteristic function of the Riemann zeta function: If n is a nearest integer to imaginary part of zero, then a(n)=1 else a(n)=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A133011 Characteristic function of the Riemann zeta function: If n is a nearest integer to imaginary part of zero, then a(n)=0 else a(n)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs