A360500 Decimal expansion of the unique positive root to zeta(s) + zeta'(s) = 0, where zeta is the Riemann zeta function and zeta' is the derivative of zeta.

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

Offset

1,2

Comments

Denote this root as s_0, then Sum_{n>=1} log(n)/n^s > Sum_{n>=1} 1/n^s if and only if 1 < s < s_0; Sum_{n>=1} log(n)/n^s < Sum_{n>=1} 1/n^s if and only if s > s_0.

Links

Table of n, a(n) for n=1..98.

Example

Uniquely for s_0 = 1.68041735920403754776..., we have Sum_{n>=1} log(n)/n^(s_0) = Sum_{n>=1} 1/n^(s_0).

Mathematica

RealDigits[s /. FindRoot[Zeta[s] + Zeta'[s] == 0, {s, 2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Jun 25 2023 *)

Prog

(PARI) solve(x=1.5, 2, zeta(x)+zeta'(x))

Crossrefs

Sequence in context: A268508 A021151 A200116 * A308314 A097909 A143819

Adjacent sequences: A360497 A360498 A360499 * A360501 A360502 A360503

Keyword

nonn,cons

Author

Jianing Song, Feb 09 2023

Status

approved