A324859 -id:A324859 - cp's OEIS frontend

A324860 Decimal expansion of 0.5250984..., a real fixed point of the iteration s = zetahurwitz(s, A324859).

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

Offset: 0

Reikku Kulon, Mar 18 2019

For real values of the parameter "a" between 0 and 1, a real fixed point "s" of the iterated Hurwitz zeta function [s = zetahurwitz(s, a)] lies on a curve that passes through A069857 (-0.295905...) and has a maximum tending toward 1. This curve has inflection points for a = 0.1990753... (A324859) or 0.91964... . The fixed point "s" on this curve for the iteration "s = zetahurwitz(s, A324859)" is 0.5250984... .

			0.525098424628892572115438912395851316429631107548...

Cf. A324859, A069857, A069995.

{ A324859 = solve(t = 1/16, 1/2, derivnum(x = t, solve(v = -1, 1 - x, v - zetahurwitz(v, x)), 2); ); solve(v = -1, 1 - A324859, v - zetahurwitz(v, A324859)) }

A069995 Decimal expansion of the real positive solution to zeta(x)=x.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, May 01 2002

Fixed point of Riemann zeta function. - Michal Paulovic, Dec 31 2017

			1.83377265168027139624564858944152359218097851880099333719403756009807267200...

G. C. Greubel, Table of n, a(n) for n = 1..10000
David Rainford, Riemann zeta function: iteration of ζ and the significance of the value 1.83377..., Prime Patterns, 2015.
David Rainford, Hurwitz zeta function: iteration fractal example near a threshold, Prime Patterns, 2019. [observed effects of tangent to fixed-point curve]

Cf. A069857, A324859, A324860.

RealDigits[ FindRoot[ Zeta[x] == x, {x, 2}, WorkingPrecision -> 2^7, AccuracyGoal -> 2^8, PrecisionGoal -> 2^7][[1, 2]], 10, 111][[1]] (* Robert G. Wilson v, Jan 07 2018 *)

solve(x=1.5,2,zeta(x)-x) \\ Michal Paulovic, Dec 31 2017

(zeta(x)==x).find_root(1,2,x) # G. C. Greubel, Apr 01 2019

Corrected and extended by Michal Paulovic, Dec 31 2017

cp's OEIS Frontend

A324860 Decimal expansion of 0.5250984..., a real fixed point of the iteration s = zetahurwitz(s, A324859).

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A069995 Decimal expansion of the real positive solution to zeta(x)=x.

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