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%I #8 Jun 25 2023 01:57:35
%S 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,
%T 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,
%U 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
%N 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.
%C 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.
%e Uniquely for s_0 = 1.68041735920403754776..., we have Sum_{n>=1} log(n)/n^(s_0) = Sum_{n>=1} 1/n^(s_0).
%t RealDigits[s /. FindRoot[Zeta[s] + Zeta'[s] == 0, {s, 2}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Jun 25 2023 *)
%o (PARI) solve(x=1.5, 2, zeta(x)+zeta'(x))
%K nonn,cons
%O 1,2
%A _Jianing Song_, Feb 09 2023
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