A344427 Decimal expansion of -zeta'(alpha), where alpha = A069995 is the fixed point of Riemann zeta function in (1, +oo).

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

Offset: 1

Author

Jianing Song, May 19 2021

Keywords

• nonn
• cons

Comments

|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.

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.

Examples

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

Crossrefs

Cf. A069995, A344428.

Programs

Mathematica

RealDigits[-Zeta'[x /. FindRoot[Zeta[x] == x, {x, 2}, WorkingPrecision -> 120]]][[1]] (* Amiram Eldar, Jun 01 2023 *)

PARI

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