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%I #18 Oct 01 2023 02:20:28
%S 6,7,0,2,5,9,7,9,8,7,6,8,5,9,9,5,0,2,8,8,3,9,1,6,4,1,1,9,6,8,6,6,7,4,
%T 4,7,4,8,0,3,9,2,7,9,0,0,9,7,4,3,4,9,1,7,3
%N Decimal expansion of a minimum of Arias de Reyna and van de Lune's kappa function.
%C The kappa function is implicitly defined by exp(2*Pi*i*kappa(t)) = -exp(-2*i*theta(t))*(zeta'(1/2-i*t)/zeta'(1/2+i*t)) and kappa(0)=-1/2.
%H J. Arias de Reyna and J. van de Lune, <a href="http://arxiv.org/abs/1305.3844">On the exact location of the non-trivial zeros of Riemann's zeta function.</a>, arXiv:1305.3844 [math.NT], 2013.
%e -0.6702597987685995028839164119686674474803927900974349173...
%t kappa[t_] := -1 - 1/Pi* Arg[ RiemannSiegelZ'[t] - I*RiemannSiegelZ[t]*RiemannSiegelTheta'[t]]; digits = 55; kappa0[n_] := kappa0[n] = FindMinimum[kappa[t], {t, 1}, WorkingPrecision -> n] [[1]] // RealDigits[#, 10, digits] & // First; kappa0[digits]; kappa0[n = 2*digits]; While[ kappa0[n] != kappa0[n - digits], n = n + digits]; kappa0[n]
%Y Cf. A114866, A225961 (position of minimum).
%K nonn,cons,more
%O 0,1
%A _Jean-François Alcover_, May 22 2013
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