A002410 -id:A002410 - cp's OEIS frontend

A122526 Complement of A002410.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 39, 40, 42, 44, 45, 46, 47, 49, 51, 52, 54, 55, 57, 58, 60, 62, 63, 64, 66, 68, 69, 71, 73, 74, 75, 78, 80, 81, 82, 84, 86, 88, 90, 91, 93, 94, 97, 98, 100, 102, 103, 106

Offset: 1

N. J. A. Sloane, Sep 18 2006

The number of terms is finite (but large).

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A002410, A120401.

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

A131582 Concatenation of first n numbers of (A002410: Nearest integer to imaginary part of n-th zero of Riemann zeta function).

Original entry on oeis.org

14, 1421, 142125, 14212530, 1421253033, 142125303338, 14212530333841, 1421253033384143, 142125303338414348, 14212530333841434850, 1421253033384143485053, 142125303338414348505356, 14212530333841434850535659, 1421253033384143485053565961

Offset: 1

Omar E. Pol, Sep 13 2007

Matthew House, Table of n, a(n) for n = 1..341
Index entries for zeta function.

See A002410 for more information.

Offset corrected by Matthew House, Nov 01 2016

A134814 Concatenation of next n members of (A002410: Nearest integer to imaginary part of n-th zero of Riemann zeta function).

Original entry on oeis.org

14, 2125, 303338, 41434850, 5356596165, 677072767779, 83858789929596, 99101104105107111112114, 116119121123124128130131133, 135138140141143146147150151153, 156158159161163166167169170173175

Offset: 1

Omar E. Pol, Nov 30 2007

Index entries for zeta function.

Cf. A002410, A131582.

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).

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[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

A058303 Decimal expansion of the imaginary part of the first nontrivial zero of the Riemann zeta function.

Original entry on oeis.org

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

Offset: 2

Robert G. Wilson v, Dec 08 2000

"The Riemann Hypothesis, considered by many to be the most important unsolved problem of mathematics, is the assertion that all of zeta's nontrivial zeros line up with the first two all of which lie on the line 1/2 + sqrt(-1)*t, which is called the critical line. It is known that the hypothesis is obeyed for the first billion and a half zeros." (Wagon)

We can compute 105 digits of this zeta zero as the numerical integral: gamma = Integral_{t=0..gamma+15} (1/2)*(1 - sign((RiemannSiegelTheta(t) + Im(log(zeta(1/2 + i*t))))/Pi - n + 3/2)) where n=1 and where the initial value of gamma = 1. The upper integration limit is arbitrary as long as it is greater than the zeta zero computed recursively. The recursive formula fails at zeta zeros with indices n equal to sequence A153815. - Mats Granvik, Feb 15 2017

			14.1347251417346937904572519835624702707842571156992...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.15.3, p. 138.
S. Wagon, "Mathematica In Action," W. H. Freeman and Company, NY, 1991, page 361.

Iain Fox, Table of n, a(n) for n = 2..20000
P. J. Forrester and A. Mays, Finite size corrections in random matrix theory and Odlyzko's data set for the Riemann zeros, arXiv preprint arXiv:1506.06531 [math-ph], 2015.
P. J. Forrester and A. Mays, Finite size corrections in random matrix theory and Odlyzko's data set for the Riemann zeros, Proceedings of the Royal Society A, Vol: 471, Issue: 2182, 2015.
Fredrik Johansson, The first nontrivial zero to over 300000 decimal digits.
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.
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros.
Eric Weisstein's World of Mathematics, Xi-Function.
Index entries for zeta function.

Imaginary part of k-th nontrivial zero of Riemann zeta function: A058303 (k=1: this), 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).

Cf. A002410 (round), A013629 (floor); A057641, A057640, A058209, A058210.

Digits:= 150; Re(fsolve(Zeta(1/2+I*t)=0, t=14.13)); # Iaroslav V. Blagouchine, Jun 24 2016

FindRoot[ Zeta[1/2 + I*t], {t, 14 + {-.3, +.3}}, AccuracyGoal -> 100, WorkingPrecision -> 120]
RealDigits[N[Im[ZetaZero[1]], 100]][[1]] (* Charles R Greathouse IV, Apr 09 2012 *)
(* The following numerical integral takes about 9 minutes to compute *)Clear[n, t, gamma]; gamma = 1; numberofzetazeros = 1; Quiet[Do[gamma = N[NIntegrate[(1/2)*(1 - Sign[(RiemannSiegelTheta[t] + Im[Log[Zeta[I*t + 1/2]]])/Pi - n + 3/2]), {t, 0, gamma + 15}, PrecisionGoal -> 110, MaxRecursion -> 350, WorkingPrecision -> 120], 105]; Print[gamma], {n, 1, numberofzetazeros}]]; RealDigits[gamma][[1]] (* Mats Granvik, Feb 15 2017 *)

solve(x=14,15,imag(zeta(1/2+x*I))) \\ Charles R Greathouse IV, Feb 26 2012

lfunzeros(1,15)[1] \\ Charles R Greathouse IV, Mar 07 2018

zeta(1/2 + i*14.1347251417346937904572519836...) = 0.

A074760 Decimal expansion of lambda(1) in Li's criterion.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Sep 28 2002

Decimal expansion of -B =(1/2)*sum(r in Z, 1/r/(1-r)) where Z is the set of zeros of the Riemann zeta function which lie in the strip 0 <= Re(z) <= 1.

According to Gun, Murty, & Rath (2018), it is not even known whether this constant is rational or not (though see Theorem 3.1), though they show that it is transcendental under Schanuel’s conjecture. - Charles R Greathouse IV, Nov 12 2021

			0.023095708966121033814310247906495291621932127152050759525392...

H. M. Edwards, Riemann's Zeta Function, Dover Publications Inc. 1974, p. 160.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.6.2, 2.21, and 2.32, pp. 42, 168, 204.
S. J. Patterson, "An introduction to the theory of the Riemann Zeta-function", Cambridge Studies in Advanced Mathematics 14, p. 34.

E. Bombieri and J. C. Lagarias, Complements to Li's Criterion for the Riemann Hypothesis, J. Number Th. 77(2) (1999), 274-287.
M. W. Coffey, Relations and positivity results for derivatives of the Riemann xi function, J. Comput. Appl. Math. 166(2) (2004), 525-534.
Sanoli Gun, M. Ram Murty, and Purusottam Rath, Transcendental sums related to the zeros of zeta functions, arXiv:1807.11201 [math.NT], 2018; Mathematika, Vol. 64, no. 3 (2018), pp. 875-897.
Xian-Jin Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Th. 65(2) (1997), 325-333.
Stephane Louboutin, Majorations explicites de |L(1, χ)| (Suite), C. R. Acad. Sci. Paris. 323, pp. 443-446 (1996). (In French). See Theorem 1 at p. 444.
J. Sondow and C. Dumitrescu, A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis, arXiv:1005.1104 [math.NT], 2010; see p. 3 in the link.
J. Sondow and C. Dumitrescu, A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis, Periodica Math. Hungarica, 60 (2010), 37-40; see p. 39 in the link.
Eric Weisstein's World of Mathematics, Li's Criterion.
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros.
Wikipedia, Li's criterion.
Index entries for zeta function.

Cf. A002410 (nearest integer to imaginary part of n-th zeta zero), A195423 (twice the constant).
Cf. A104539 (lambda_2), A104540 (lambda_3), A104541 (lambda_4), A104542 (lambda_5).
Cf. A306339 (lambda_6), A306340 (lambda_7), A306341 (lambda_8).

RealDigits[EulerGamma/2 + 1 - Log[4 Pi]/2, 10, 110][[1]]

Euler/2+1-log(4*Pi)/2 \\ Charles R Greathouse IV, Jan 26 2012

-B = Gamma/2 + 1 - log(4*Pi)/2 = 0.0230957...

Name simplified by Eric W. Weisstein, Feb 08 2019

A065434 Decimal expansion of imaginary part of 2nd nontrivial zero of Riemann zeta function.

Original entry on oeis.org

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

Offset: 2

N. J. A. Sloane, Nov 24 2001

			The zero is at 1/2 + i*21.0220396387715549926284795938969...

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

Enrico Bombieri, Problems of the Millennium: the Riemann Hypothesis, Clay Mathematics Institute.
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: this), A065452 (k=3), A065453 (k=4), A192492 (k=5), A305741 (k=6), A305742 (k=7), A305743 (k=8), A305744 (k=9), A306004 (k=10).

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

Digits:= 150; Re(fsolve(Zeta(1/2+I*t)=0, t=21)); # Iaroslav V. Blagouchine, Jun 25 2016

ZetaZero[2] // Im // RealDigits[#, 10, 99]& // First (* Jean-François Alcover, Mar 05 2013 *)

solve(x=21,22,real(zeta(1/2+x*I))) \\ Charles R Greathouse IV, Jun 30 2011

lfunzeros(1,[21,22])[1] \\ M. F. Hasler, Nov 23 2018

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

cp's OEIS Frontend

A122526 Complement of A002410.

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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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A131582 Concatenation of first n numbers of (A002410: Nearest integer to imaginary part of n-th zero of Riemann zeta function).

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A134814 Concatenation of next n members of (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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A058303 Decimal expansion of the imaginary part of the first nontrivial zero of the Riemann zeta function.

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A074760 Decimal expansion of lambda(1) in Li's criterion.

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