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A387052 Decimal expansion of -x, where x is the abscissa of the second local extremum of the Riemann zeta function on the negative real axis.

%I A387052 #30 Aug 24 2025 16:23:48
%S A387052 4,9,3,6,7,6,2,1,0,8,5,9,4,9,4,7,8,6,8,8,7,9,3,5,8,2,4,9,8,4,2,7,1,5,
%T A387052 3,7,3,6,6,1,0,0,9,2,0,3,5,0,5,7,5,5,6,2,2,2,9,5,6,3,3,3,4,2,0,4,4,9,
%U A387052 4,2,0,2,9,1,1,9,8,2,4,3,7,4,2,0,3,7,0,2,2,2,6,9,6,6,7,9,3,7,8,8,6,9,8,9,9
%N A387052 Decimal expansion of -x, where x is the abscissa of the second local extremum of the Riemann zeta function on the negative real axis.
%C A387052 The Riemann zeta function has zeros for x = -2*n for n >= 1, which means that between -2*(n+1) and -2*n the function has an extremum for each positive integer n.
%C A387052 For the value of zeta(-x_2) see A387065.
%C A387052 It is an open question whether the fractional part of x_n tends to 1 or some unknown constant c < 1 as n tends to infinity.
%H A387052 Artur Jasinski, <a href="/A387052/a387052_1.txt">Complete list of the first 1000 negative extrema of the Riemann zeta function</a>.
%t A387052 kk = x /. FindRoot[Zeta'[x] == 0, {x, -5}, WorkingPrecision -> 110];
%t A387052 RealDigits[kk, 10, 105][[1]]
%o A387052 (PARI) solve(x=-4.95, -4.9, zeta'(x))
%Y A387052 Cf. A271855, A271856.
%K A387052 nonn,cons,new
%O A387052 1,1
%A A387052 _Artur Jasinski_, Aug 15 2025