A344427 - OEIS

%I #12 Jun 01 2023 01:58:50

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

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

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

%N Decimal expansion of -zeta'(alpha), where alpha = A069995 is the fixed point of Riemann zeta function in (1, +oo).

%C |zeta'(alpha)| > 1 means that s = alpha is a repelling fixed point of zeta(s). As a result, for any initial value s_0 in (1, +oo), s_0 != alpha, the iterated sequence s_0, zeta(s_0), zeta(zeta(s_0)), ... diverges.

%C Moreover, let s_0 be any real number > alpha, s_n = zeta(s_{n-1}) for n >= 1, then it seems that ... > s_{2n} > s_{2n-2} > ... > s_2 > s_0 > alpha > s_1 > s_3 > ... > s_{2n+1} > ..., and {s_{2n}} diverges to +oo, {s_{2n+1}} converges to 1. Moreover, the divergence of {s_{2n}} and convergence of {s_{2n+1}} should be really fast, see my conjecture in A344428.

%e zeta'(1.83377265168027139624...) = -1.37425243024718990618...

%t RealDigits[-Zeta'[x /. FindRoot[Zeta[x] == x, {x, 2}, WorkingPrecision -> 120]]][[1]] (* _Amiram Eldar_, Jun 01 2023 *)

%o (PARI) default(realprecision, 100); zeta'(solve(x=1.5, 2, zeta(x)-x))

%Y Cf. A069995, A344428.

%K nonn,cons

%O 1,2

%A _Jianing Song_, May 19 2021