A065434 -id:A065434 - 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

A123506 Sequence generated from the second nontrivial zero of the Riemann zeta function.

Original entry on oeis.org

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

Offset: 2

Gary W. Adamson, Oct 02 2006

nonn

A123504 performs an analogous set of operations using the first nontrivial zero. A123507 records the lengths of runs in this sequence.

Let z = (1/2 + i*t), t = 21.0220396387... (the second nontrivial Riemann zeta function zero). Perform (1/n)^z, (n = 2, 3, 4, ...) extracting the argument. If the argument is between 0 and 180 degrees, a(n) = 1, otherwise a(n) = 0.

			a(7) = 1 since (1/7)^z = (0.37796447..., angle 176.201... degrees) and the argument is between 0 and 180 degrees.

John Derbyshire, Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Plume - a Penguin Group, NY, 2003, pp. 198-199.

Cf. A065434, A102522, A102523, A123504, A123505, A123507.

a[n_] := Boole[Arg[1/n^ZetaZero[2]] > 0]; Array[a, 100, 2] (* Amiram Eldar, May 31 2025 *)

More terms from Amiram Eldar, May 31 2025

A131584 Concatenation of first n numbers of the decimal expansion of imaginary part of 2nd nontrivial zero of Riemann zeta function.

Original entry on oeis.org

2, 21, 210, 2102, 21022, 210220, 2102203, 21022039, 210220396, 2102203963, 21022039638, 210220396387, 2102203963877, 21022039638771, 210220396387715, 2102203963877155, 21022039638771554, 210220396387715549

Offset: 0

Omar E. Pol, Sep 13 2007

Index entries for zeta function.

See A002410 and A065434 for more information.

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.

A199501 Pi raised to fraction of imaginary part of second and third Riemann zeta function zero.

Original entry on oeis.org

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

Offset: 1

Mats Granvik, Nov 07 2011

This number has seven consecutive 7:s in its decimal expansion close to the beginning. This can be amplified by dividing the number by 589, which gives 0.004443727297444444444471198171188... which in turn has ten consecutive 4:s.

			2.61735537819477777779353572283019080292...

Cf. A000796, A065434, A065452.

RealDigits[N[Pi^(Im[ZetaZero[2]]/Im[ZetaZero[3]]), 90]]

This number = A000796^(A065434/A065452)

A346289 Decimal expansion of the length of first, and largest, gap between nontrivial zeta zeros.

Original entry on oeis.org

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

Offset: 1

Charles R Greathouse IV, Jul 13 2021

Simonič shows that this gap is in fact larger than all other gaps between zeros of the Riemann zeta function, see the proof of Lemma 3.

			6.8873144970368612021712276103344325065500834092035385789439529434376351562596....

Aleksander Simonič, Lehmer pairs and derivatives of Hardy's Z-function, arXiv:1612.08627 [math.NT], 2016-2017.
Index entries for zeta function.

Cf. A065434, A058303.

Im[ZetaZero[2] - ZetaZero[1]] // RealDigits[#, 10, 88]& // First (* Peter Luschny, Jul 16 2021 *)

PARI
```
call((x,y)->y-x,lfunzeros(1,22))
```

Equals A065434 - A058303.

cp's OEIS Frontend

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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A123506 Sequence generated from the second nontrivial zero of the Riemann zeta function.

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A131584 Concatenation of first n numbers of the decimal expansion of imaginary part of 2nd nontrivial zero of Riemann zeta function.

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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.

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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.

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Comments

Examples

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A199501 Pi raised to fraction of imaginary part of second and third Riemann zeta function zero.

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Formula

A346289 Decimal expansion of the length of first, and largest, gap between nontrivial zeta zeros.

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