cp's OEIS Frontend

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

%I A058303 #75 Feb 16 2025 08:32:43
%S A058303 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,
%T A058303 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,
%U A058303 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
%N A058303 Decimal expansion of the imaginary part of the first nontrivial zero of the Riemann zeta function.
%C A058303 "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)
%C A058303 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
%D A058303 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.15.3, p. 138.
%D A058303 S. Wagon, "Mathematica In Action," W. H. Freeman and Company, NY, 1991, page 361.
%H A058303 Iain Fox, <a href="/A058303/b058303.txt">Table of n, a(n) for n = 2..20000</a>
%H A058303 P. J. Forrester and A. Mays, <a href="http://arxiv.org/abs/1506.06531">Finite size corrections in random matrix theory and Odlyzko's data set for the Riemann zeros</a>, arXiv preprint arXiv:1506.06531 [math-ph], 2015.
%H A058303 P. J. Forrester and A. Mays, <a href="https://doi.org/10.1098/rspa.2015.0436">Finite size corrections in random matrix theory and Odlyzko's data set for the Riemann zeros</a>, Proceedings of the Royal Society A, Vol: 471, Issue: 2182, 2015.
%H A058303 Fredrik Johansson, <a href="http://fredrikj.net/math/rho1_300k_decimal.txt">The first nontrivial zero to over 300000 decimal digits</a>.
%H A058303 Andrew M. Odlyzko, <a href="http://www.plouffe.fr/simon/constants/zeta100.html">The first 100 (non trivial) zeros of the Riemann Zeta function, to over 1000 decimal digits each</a>, AT&T Labs - Research.
%H A058303 Andrew M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html">Tables of zeros of the Riemann zeta function</a>.
%H A058303 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RiemannZetaFunctionZeros.html">Riemann Zeta Function Zeros</a>.
%H A058303 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Xi-Function.html">Xi-Function</a>.
%H A058303 <a href="/index/Z#zeta_function">Index entries for zeta function</a>.
%F A058303 zeta(1/2 + i*14.1347251417346937904572519836...) = 0.
%e A058303 14.1347251417346937904572519835624702707842571156992...
%p A058303 Digits:= 150; Re(fsolve(Zeta(1/2+I*t)=0, t=14.13)); # _Iaroslav V. Blagouchine_, Jun 24 2016
%t A058303 FindRoot[ Zeta[1/2 + I*t], {t, 14 + {-.3, +.3}}, AccuracyGoal -> 100, WorkingPrecision -> 120]
%t A058303 RealDigits[N[Im[ZetaZero[1]], 100]][[1]] (* _Charles R Greathouse IV_, Apr 09 2012 *)
%t A058303 (* 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 *)
%o A058303 (PARI) solve(x=14,15,imag(zeta(1/2+x*I))) \\ _Charles R Greathouse IV_, Feb 26 2012
%o A058303 (PARI) lfunzeros(1,15)[1] \\ _Charles R Greathouse IV_, Mar 07 2018
%Y A058303 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).
%Y A058303 Cf. A002410 (round), A013629 (floor); A057641, A057640, A058209, A058210.
%K A058303 nonn,cons,easy
%O A058303 2,2
%A A058303 _Robert G. Wilson v_, Dec 08 2000