A144738 - OEIS

%I #23 Apr 29 2023 00:05:59

%S 5,1,2,7,3,7,9,1,5,4,5,4,9,6,9,3,3,5,3,2,9,2,2,7,0,9,9,7,0,6,0,7,5,2,

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

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

%N Decimal expansion of constant related to a dynamical system involving the zeta function.

%C If iterations of zeta function converge to the constant A069857 then the ratio of successive imaginary parts of the orbit converge to -c. I.e., let z(n+1) = zeta(z(n)) if lim_{n->oo} z(n) = A069857; then lim_{n->oo} imag(z(n+1))/imag(z(n)) = -0.512....

%C -c = zeta'(A069857). - _Gerald McGarvey_, Feb 22 2009

%H B. Brent, <a href="http://barrybrent.9f.com/storage/my%20math/zeta.pdf">Experiments with zeta dynamics</a>, Feb 2012. [Broken link]

%H B. Brent, <a href="https://arxiv.org/abs/1703.08779">Experiments with the dynamics of the Riemann zeta function</a>, arXiv:1703.08779 [math.NT], 2017.

%e c=0.51273791545496933532922709970607529512404828484563...

%t digits = 104; A069857 = x /. FindRoot[ Zeta[x] == x, {x, 0}, WorkingPrecision -> digits+5]; Zeta'[A069857] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Mar 04 2013, after _Gerald McGarvey_ *)

%o (PARI) -zeta'(solve(x=-1, 0, zeta(x)-x)) \\ _Michel Marcus_, May 05 2020

%Y Cf. A069857.

%K cons,nonn

%O 0,1

%A _Benoit Cloitre_, Sep 20 2008