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

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

2,2

Comments

"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

References

S. Wagon, "Mathematica In Action," W. H. Freeman and Company, NY, 1991, page 361.

Links

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.

Formula

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

Example

14.1347251417346937904572519835624702707842571156992...

Maple

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

Mathematica

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 *)

Prog

(PARI) solve(x=14, 15, imag(zeta(1/2+x*I))) \\ Charles R Greathouse IV, Feb 26 2012
(PARI) lfunzeros(1, 15)[1] \\ Charles R Greathouse IV, Mar 07 2018

Crossrefs

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.

Sequence in context: A376312 A362331 A376264 * A240935 A090724 A343571

Adjacent sequences: A058300 A058301 A058302 * A058304 A058305 A058306

Keyword

nonn,cons,easy

Author

Robert G. Wilson v, Dec 08 2000

Status

approved