A069995 - OEIS

%I #26 May 09 2019 10:40:13

%S 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,

%T 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,

%U 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

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

%C Fixed point of Riemann zeta function. - _Michal Paulovic_, Dec 31 2017

%H G. C. Greubel, <a href="/A069995/b069995.txt">Table of n, a(n) for n = 1..10000</a>

%H David Rainford, <a href="http://primepatterns.tumblr.com/post/134491513696/riemann-zeta-function-iteration-of-%CE%B6-and-the">Riemann zeta function: iteration of ζ and the significance of the value 1.83377...</a>, Prime Patterns, 2015.

%H David Rainford, <a href="http://primepatterns.tumblr.com/post/182029767271/hurwitz-zeta-function-iteration-fractal-example">Hurwitz zeta function: iteration fractal example near a threshold</a>, Prime Patterns, 2019. [observed effects of tangent to fixed-point curve]

%e 1.83377265168027139624564858944152359218097851880099333719403756009807267200...

%t 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 *)

%o (PARI) solve(x=1.5,2,zeta(x)-x) \\ _Michal Paulovic_, Dec 31 2017

%o (Sage) (zeta(x)==x).find_root(1,2,x) # _G. C. Greubel_, Apr 01 2019

%Y Cf. A069857, A324859, A324860.

%K cons,easy,nonn

%O 1,2

%A _Benoit Cloitre_, May 01 2002

%E Corrected and extended by _Michal Paulovic_, Dec 31 2017