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A271855 Decimal expansion of -x_1 such that the Riemann function zeta(x) has at real x_1<0 its first local extremum.

%I A271855 #12 Feb 16 2025 08:33:33
%S A271855 2,7,1,7,2,6,2,8,2,9,2,0,4,5,7,4,1,0,1,5,7,0,5,8,0,6,6,1,6,7,6,5,2,8,
%T A271855 4,1,2,4,2,4,7,5,1,8,5,3,9,1,7,4,9,2,6,5,5,9,4,4,0,7,2,7,5,9,7,2,9,0,
%U A271855 3,9,8,3,2,6,1,3,9,3,0,8,7,8,2,7,6,7,1,2,1,1,4,4,2,6,1,6,8,9,1,9,8,4,5,3,6
%N A271855 Decimal expansion of -x_1 such that the Riemann function zeta(x) has at real x_1<0 its first local extremum.
%C A271855 For real x < 0, zeta(x) undergoes divergent oscillations, passing through zero at every even integer value of x. In each interval (-2n,-2n-2), n = 1, 2, 3, ..., it attains a local extreme (maximum, minimum, maximum, ...). The location x_n of the n-th local extreme does not match the odd integer -2n-1. Rather, x_n > -2n-1 for n = 1 and 2, and x_n < -2n-1 for n >= 3. This entry defines the location x_1 of the first maximum. The corresponding value is in A271856.
%H A271855 Stanislav Sykora, <a href="/A271855/b271855.txt">Table of n, a(n) for n = 1..2000</a>
%H A271855 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RiemannZetaFunction.html">Riemann Zeta Function</a>
%e A271855 x_1 = -2.7172628292045741015705806616765284124247518539174926559440...
%e A271855 zeta(x_1) = A271856.
%o A271855 (PARI) \\ This function was tested up to n = 11600000:
%o A271855 zetaextreme(n) = {solve(x=-2.0*n,-2.0*n-1.9999999999,zeta'(x))}
%o A271855 a = -zetaextreme(1) \\ Evaluation for this entry
%Y A271855 Cf. A271856.
%K A271855 nonn,cons
%O A271855 1,1
%A A271855 _Stanislav Sykora_, Apr 23 2016