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

Author

Benoit Cloitre, Sep 20 2008

Keywords

• cons
• nonn

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

Examples

			c=0.51273791545496933532922709970607529512404828484563...
		

Links

• 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.

Crossrefs

Cf. A069857.

Programs

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 *)

PARI

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