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A344427 Decimal expansion of -zeta'(alpha), where alpha = A069995 is the fixed point of Riemann zeta function in (1, +oo). 1
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..90.

EXAMPLE

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

PROG

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

CROSSREFS

Cf. A069995, A344428.

Sequence in context: A077226 A175316 A197145 * A163335 A266273 A341605

Adjacent sequences:  A344424 A344425 A344426 * A344428 A344429 A344430

KEYWORD

nonn,cons

AUTHOR

Jianing Song, May 19 2021

STATUS

approved

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Last modified September 23 14:09 EDT 2021. Contains 347617 sequences. (Running on oeis4.)