A069857 -id:A069857 - cp's OEIS frontend

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

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

A324859 Decimal expansion of 0.1990753..., an inflection point of a Hurwitz zeta fixed-point function.

Original entry on oeis.org

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

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... or 0.91964... . The fixed point "s" on this curve for the iteration "s = zetahurwitz(s, A324859)" is A324860 (0.5250984...).

			0.1990753035447728549711300350722284216882866320163...

Reikku Kulon, Plot of Hurwitz zeta fixed-point curve for 0 < a < 2 and -1 < s < +1.

Cf. A069857, A069995, A324860.

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

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

Original entry on oeis.org

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)) }

A072115 Continued fraction expansion of abs(C) where C=-0.2959050055752...is the real negative solution to zeta(x)=x.

Original entry on oeis.org

0, 3, 2, 1, 1, 1, 2, 1, 7, 14, 1, 2, 10, 1, 5, 3, 1, 7, 2, 1, 2, 2, 2, 4, 1, 1, 12, 1, 1, 1, 14, 2, 10, 3, 5, 6, 2, 1, 6, 13, 1, 2, 2, 4, 8, 1, 4, 8, 2, 1, 16, 1, 1, 1, 1, 4, 2, 1, 1, 1, 3, 13, 4, 1, 2, 1, 6, 1, 1, 2, 43, 1, 3, 1, 1, 2, 2, 2, 1, 2, 2, 2, 10, 5, 4, 8, 1, 5, 3, 2, 1, 1, 3, 2, 19

Offset: 0

Benoit Cloitre, Jun 19 2002

Start from any complex number z=x+iy, not solution to zeta(z)=z, iterate the zeta function on z. If zeta_m(z) (=zeta(zeta(....(z)..)) m times) has a limit when m grows, then this limit seems to always be the real number : C=-0.2959050055752....Example: if z=3+5I after 30 iterations : zeta_30(z)=-0.29590556499...-0.00000041029065...*I

Cf. A069857 (decimal expansion).

\p150 contfrac(abs(solve(X=-1,0,zeta(X)-X)))

Offset changed by Andrew Howroyd, Jul 06 2024

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

Original entry on oeis.org

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

Benoit Cloitre, Sep 20 2008

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

			c=0.51273791545496933532922709970607529512404828484563...

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.

Cf. A069857.

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 *)

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

A307065 Decimal expansion of the negative real attracting fixed point of Э(s) = (1 - 2^s) * (1 - 2^(1 - s)) * gamma(s) * zeta(s) * beta(s) / Pi^s.

Original entry on oeis.org

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

Offset: 0

Reikku Kulon, Mar 22 2019

Ossicini's function Э(s) is constructed to remove the poles of gamma(s) and zeta(s) along with the trivial zeros of zeta(s) and (Dirichlet) beta(s). Its zeros include the nontrivial zeros of zeta(s) and beta(s), and complex zeros contributed by (1 - 2^s) and (1 - 2^(1 - s)) at regular intervals of 2*Pi/log(2) on the lines Re(s) = {0, 1}.

			-0.1784830971429545702860575466420370769978315915595...

A. Ossicini, An alternative form of the functional equation for Riemann's Zeta function, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 56 (2008/09), 95-111.

Andrea Ossicini, An Alternative Form of the Functional Equation for Riemann's Zeta Function, II, arXiv:1206.4494 [math.HO], 2012-2014.

Cf. A069857, A069995.

f[s_] := s - (1 - 2^s)(1 - 2^(1-s)) Gamma[s] Zeta[s] ((HurwitzZeta[s, 1/4] - HurwitzZeta[s, 3/4])/(4 Pi)^s);
s0 = s /. FindRoot[f[s], {s, -1/5}, WorkingPrecision -> 100];
RealDigits[s0][[1]] (* Jean-François Alcover, May 07 2019 *)

solve(s = -1/2, -1/8, s - (1 - 2^s) * (1 - 2^(1 - s)) * gamma(s) * zeta(s) * (zetahurwitz(s, 1/4) - zetahurwitz(s, 3/4)) / (4 * Pi)^s)

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

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A324859 Decimal expansion of 0.1990753..., an inflection point of a Hurwitz zeta fixed-point function.

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

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A072115 Continued fraction expansion of abs(C) where C=-0.2959050055752...is the real negative solution to zeta(x)=x.

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

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A307065 Decimal expansion of the negative real attracting fixed point of Э(s) = (1 - 2^s) * (1 - 2^(1 - s)) * gamma(s) * zeta(s) * beta(s) / Pi^s.

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