A092783 -id:A092783 - cp's OEIS frontend

A002410 Nearest integer to imaginary part of n-th zero of Riemann zeta function.

14, 21, 25, 30, 33, 38, 41, 43, 48, 50, 53, 56, 59, 61, 65, 67, 70, 72, 76, 77, 79, 83, 85, 87, 89, 92, 95, 96, 99, 101, 104, 105, 107, 111, 112, 114, 116, 119, 121, 123, 124, 128, 130, 131, 133, 135, 138, 140, 141, 143, 146, 147, 150, 151, 153, 156, 158, 159, 161

Offset: 1

N. J. A. Sloane

"All these zeros of the form s + it have real part s = 1/2 and are simple. Thus the Riemann hypothesis is true at least for t < 3330657430697." - Wedeniwski
From Daniel Forgues, Jul 24 2009: (Start)
All nontrivial zeros on the critical line, of the form 1/2 + i*t, have an associated conjugate nontrivial zero of the form 1/2 - i*t.
Any nontrivial zeros off the critical line, if ever found, would come in pairs (1/2 +- delta) + i*t, 0 < delta < 1/2. Each of these pairs, again if ever found, would then have their associated conjugate pair (1/2 +- delta) - i*t, 0 < delta < 1/2. (End)
The sequence is not strictly increasing. - Joerg Arndt, Jan 17 2015
The fraction of numbers n such that a(n) = a(n-1) has density 1. There are only finitely many numbers n with a(n) > a(n-1) + 1, see A208436. - Charles R Greathouse IV, Mar 07 2018
Conjecture: Noninteger rationals of the form m/2^bigomega(m) that can be used to approximate this sequence, i.e. a(n) ~~ 2*Pi*A374074(n)/2^bigomega(A374074(n)) - n/2 +- (...), where '~~' means 'close to'. - Friedjof Tellkamp, Jul 04 2024

			The imaginary parts of the first 4 zeros are 14.134725... (A058303), 21.0220396... (A065434), 25.01085758... (A065452), 30.424876... (A065453).

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W. F. Galway, Computations related to the Riemann Hypothesis
R. Garunkstis and J. Steuding, On the distribution of zeros of the Hurwitz zeta-function, Math. Comput. 76 (2007), 323-337.
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D. A. Goldston & S. M. Gonek, A note on S(T) and the zeros of the Riemann zeta-function, arXiv:math/0511092 [math.NT], 2005.
S. Gonek, Three Lectures on the Riemann Zeta-Function, arXiv:math/0401126 [math.NT], 2004.
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Mats Granvik, Illustration of Andre LeClair's approximation of the initial terms
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E. Klarreich, Prime Time
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A. LeClair, An electrostatic depiction of the validity of the Riemann Hypothesis and a formula for the N-th zero at large N, arXiv:1305.2613 [math-ph], 2013.
N. Levinson, At Least One-Third of Zeros of Riemann's Zeta-Function are on sigma=1/2, Proc Natl Acad Sci U S A. 1974 Apr; 71(4): 1013-1015.
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A. M. Odlyzko & H. J. J. te Riele, Disproof of the Mertens Conjecture
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J. C. Perez-Moure, Evidences that the Riemann Hypothesis is true
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H. J. J. te Riele, Some historical and other notes about the Mertens conjecture and its recent disproof
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Index entries for zeta function

Cf. A013629 (floor), A092783 (ceiling), A057641, A057640, A058209, A058210, A120401, A122526, A072080, A124288 ("unstable" zeta zeros), A124289 ("unstable twins"), A236212, A177885, A374074 (approximation).

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), A306004 (k=10).

Table[Round[Im[ZetaZero[n]]], {n, 59}] (* Jean-François Alcover, May 02 2011 *)

apply(round,lfunzeros(lzeta,100)) \\ Charles R Greathouse IV, Mar 10 2016

def A002410_list(n):
    Z = lcalc.zeros(n)
    return [round(z) for z in Z]
A002410_list(59) # Peter Luschny, May 02 2014

a(n) ~ (2*Pi*e) * e^(W0(n/e)), where W0 is the principal branch of Lambert's W function. - Charles R Greathouse IV, Sep 14 2012, corrected by Hal M. Switkay, Oct 04 2021

a(n) ~ 2*Pi*(n - 11/8)/ProductLog((n - 11/8)/exp(1)). This is the asymptotic by Guilherme França and André LeClair. - Mats Granvik, Mar 10 2015; corrected May 16 2016

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 08 2004

A013629 Floor of imaginary parts of nontrivial zeros of Riemann zeta function.

Original entry on oeis.org

14, 21, 25, 30, 32, 37, 40, 43, 48, 49, 52, 56, 59, 60, 65, 67, 69, 72, 75, 77, 79, 82, 84, 87, 88, 92, 94, 95, 98, 101, 103, 105, 107, 111, 111, 114, 116, 118, 121, 122, 124, 127, 129, 131, 133, 134, 138, 139, 141, 143, 146, 147, 150, 150, 153, 156, 157, 158, 161

Offset: 1

John Morrison (John.Morrison(AT)armltd.co.uk)

nonn

			The imaginary parts of the first 4 zeros are 14.134725... (A058303), 21.0220396... (A065434), 25.01085758... (A065452), 30.424876... (A065453). Therefore the sequence starts: 14, 21, 25, 30, ..., as does A002410 (rounded values; main entry). But the 5th, 6th and 7th values are 32.935... (A192492), 37.586... (A305741), 40.9187... (A305742), whence a(n) = A002410(n)-1 and A002410 = A092783 (ceiling) for these. - _M. F. Hasler_, Nov 23 2018

H. M. Edwards, Riemann's Zeta Function, Academic Press, NY, 1974, p. 96.
C. B. Haselgrove and J. C. P. Miller, Tables of the Riemann Zeta Function. Royal Society Mathematical Tables, Vol. 6, Cambridge Univ. Press, 1960, p. 58.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for zeta function.

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

Table[Floor[Im[ZetaZero[n]]], {n, 60}] (* Alonso del Arte, Feb 07 2011 *)

lfunzeros(lzeta,100)\1 \\ Charles R Greathouse IV, Mar 10 2016

def A013629_list(n):
    Z = lcalc.zeros(n)
    return [floor(z) for z in Z]
A013629_list(50) # Peter Luschny, May 02 2014

a(n) ~ 2*Pi*n/log n. - Charles R Greathouse IV, Jun 30 2011
a(n) ~ (2*Pi*e) * e^(W0(n/e)), where W0 is the principal branch of Lambert's W function. - Hal M. Switkay, Oct 04 2021
a(n) = A092783(n) - 1. - M. F. Hasler, Nov 23 2018

Edited by Daniel Forgues, Jun 30 2011

Definition corrected by Jonathan Sondow, Sep 18 2011

A065452 Decimal expansion of imaginary part of 3rd nontrivial zero of Riemann zeta function.

Original entry on oeis.org

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

Offset: 2

N. J. A. Sloane, Nov 24 2001

See A002410 and A058303 for more information.

			The zero is at 1/2 + i * 25.01085758014568876321379099256282181865954967...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.15.3, p. 138.

G. C. Greubel, Table of n, a(n) for n = 2..10000
Andrew M. Odlyzko, The first 100 (non trivial) zeros of the Riemann Zeta function, to over 1000 decimal digits each, AT&T Labs - Research.
Andrew M. Odlyzko, Tables of zeros of the Riemann zeta function
Index entries for zeta function.

Imaginary part of k-th nontrivial zero of Riemann zeta function: A058303 (k=1), A065434 (k=2), A065452 (k=3: this), A065453 (k=4), A192492 (k=5), A305741 (k=6), A305742 (k=7), A305743 (k=8), A305744 (k=9), A306004 (k=10).

Cf. A002410 (these values rounded to nearest integer), A013629 (floor), A092783 (ceiling).

ZetaZero[3] // Im // RealDigits[#, 10, 99]& // First (* Jean-François Alcover, Mar 05 2013 *)

solve(x=25,25.1,real(zeta(1/2+x*I))) \\ Charles R Greathouse IV, Jun 30 2011

lfunzeros(1,[25,26])[1] \\ or: lfunzeros(1,26)[3]. - M. F. Hasler, Nov 23 2018

Minor edits by M. F. Hasler, Nov 23 2018

A065453 Decimal expansion of imaginary part of 4th nontrivial zero of Riemann zeta function.

Original entry on oeis.org

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

Offset: 2

N. J. A. Sloane, Nov 24 2001

See A002410 and A058303 for more information.

			The zero is at 1/2 + i * 30.42487612585951321031189753058409132...

G. C. Greubel, Table of n, a(n) for n = 2..10000
Andrew M. Odlyzko, The first 100 (non trivial) zeros of the Riemann Zeta function, to over 1000 decimal digits each, AT&T Labs - Research.
Andrew M. Odlyzko, Tables of zeros of the Riemann zeta function
Index entries for 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: this), A192492 (k=5), A305741 (k=6), A305742 (k=7), A305743 (k=8), A305744 (k=9), A306004 (k=10).

Cf. A002410 (round), A013629 (floor), A092783 (ceiling).

RealDigits[ Im[ ZetaZero[4]], 10, 99] // First (* Jean-François Alcover, Mar 07 2013 *)

solve(x=30,31,real(zeta(1/2+x*I))) \\ Charles R Greathouse IV, Mar 10 2016

lfunzeros(lzeta,[30,31])[1] \\ Charles R Greathouse IV, Mar 10 2016

A192492 Decimal expansion of imaginary part of 5th nontrivial zero of Riemann zeta function.

Original entry on oeis.org

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

Offset: 2

Alonso del Arte, Jul 02 2011

The real part of the 5th nontrivial zero is of course 1/2 (A020761; the Riemann hypothesis is here assumed to be true).

			The zero is at 1/2 + i * 32.9350615877391896906623689640749...

Andrew M. Odlyzko, The first 100 (non trivial) zeros of the Riemann Zeta function, to over 1000 decimal digits each, AT&T Labs - Research.
Andrew M. Odlyzko, Tables of zeros of the Riemann zeta function

Cf. A002410: nearest integer to imaginary part of n-th zero of Riemann zeta function (main entry); also A013629 (floor) and A092783 (ceiling).
The imaginary parts of the first 4 zeros are 14.134725... (A058303), 21.0220396... (A065434), 25.01085758... (A065452), 30.424876... (A065453). Others are A305741 (k=6), A305742 (k=7), A305743 (k=8), A305744 (k=9), A306004 (k=10).
The real parts of the trivial zeros are given by A005843 multiplied by -1 (and ignoring the initial 0 of that sequence).

(* ZetaZero was introduced in Version 6.0 *) RealDigits[ZetaZero[5], 10, 100][[1]]

solve(y=32,33,real(zeta(1/2+y*I))) \\ Charles R Greathouse IV, Mar 10 2016

lfunzeros(lzeta,[32,33])[1] \\ Charles R Greathouse IV, Mar 10 2016

Example and cross-references edited by M. F. Hasler, Nov 23 2018

A305741 Decimal expansion of imaginary part of 6th nontrivial zero of Riemann zeta function.

Original entry on oeis.org

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

Offset: 2

Seiichi Manyama, Jun 23 2018

			The zero is at 1/2 + i * 37.58617815882567125721776348070533282140559735083...

Andrew M. Odlyzko, Tables of zeros of the Riemann 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), this sequence (k=6), A305742 (k=7), A305743 (k=8), A305744 (k=9), A306004 (k=10).

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

RealDigits[Im[ZetaZero[6]], 10, 120][[1]] (* Vaclav Kotesovec, Jun 23 2018 *)

lfunzeros(1,[37,38])[1] \\ M. F. Hasler, Nov 23 2018

A305742 Decimal expansion of imaginary part of 7th nontrivial zero of Riemann zeta function.

Original entry on oeis.org

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

Offset: 2

Seiichi Manyama, Jun 23 2018

			The zero is at 1/2 + i * 40.918719012147495187398126914633254395726165962777...

Andrew M. Odlyzko, Tables of zeros of the Riemann 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), this sequence (k=7), A305743 (k=8), A305744 (k=9), A306004 (k=10).

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

RealDigits[Im[ZetaZero[7]], 10, 120][[1]] (* Vaclav Kotesovec, Jun 23 2018 *)

lfunzeros(1,[40,41])[1] \\ M. F. Hasler, Nov 23 2018

A305743 Decimal expansion of imaginary part of 8th nontrivial zero of Riemann zeta function.

Original entry on oeis.org

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

Offset: 2

Seiichi Manyama, Jun 23 2018

			The zero is at 1/2 + I*43.3270732809149995194961221654068... - _M. F. Hasler_, Nov 21 2018

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), this sequence (k=8), A305744 (k=9), A306004 (k=10).

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

RealDigits[Im[ZetaZero[8]], 10, 120][[1]] (* Vaclav Kotesovec, Jun 23 2018 *)

solve(X=43,44,imag(zeta(0.5+X*I))) \\ M. F. Hasler, Nov 21 2018

lfunzeros(1,[43,44])[1] \\ M. F. Hasler, Nov 23 2018

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

Original entry on oeis.org

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.

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

cp's OEIS Frontend

A002410 Nearest integer to imaginary part of n-th zero of Riemann zeta function.

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A013629 Floor of imaginary parts of nontrivial zeros of Riemann zeta function.

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A065452 Decimal expansion of imaginary part of 3rd nontrivial zero of Riemann zeta function.

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A065453 Decimal expansion of imaginary part of 4th nontrivial zero of Riemann zeta function.

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A192492 Decimal expansion of imaginary part of 5th nontrivial zero of Riemann zeta function.

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A305741 Decimal expansion of imaginary part of 6th nontrivial zero of Riemann zeta function.

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A305742 Decimal expansion of imaginary part of 7th nontrivial zero of Riemann zeta function.