A144738 Decimal expansion of constant related to a dynamical system involving the zeta function.

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, 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, 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

Offset

0,1

Comments

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 = zeta'(A069857). - Gerald McGarvey, Feb 22 2009

Links

Table of n, a(n) for n=0..103.

B. Brent, Experiments with zeta dynamics, Feb 2012. [Broken link]

B. Brent, Experiments with the dynamics of the Riemann zeta function, arXiv:1703.08779 [math.NT], 2017.

Example

c=0.51273791545496933532922709970607529512404828484563...

Mathematica

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 *)

Prog

(PARI) -zeta'(solve(x=-1, 0, zeta(x)-x)) \\ Michel Marcus, May 05 2020

Crossrefs

Cf. A069857.

Sequence in context: A195407 A011509 A361970 * A371849 A021199 A073324

Adjacent sequences: A144735 A144736 A144737 * A144739 A144740 A144741

Keyword

cons,nonn

Author

Benoit Cloitre, Sep 20 2008

Status

approved