OFFSET
0,1
COMMENTS
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.
LINKS
J. Arias de Reyna and J. van de Lune, On the exact location of the non-trivial zeros of Riemann's zeta function, arXiv:1305.3844 [math.NT], 2013.
EXAMPLE
0.779853575338836030518209208122537107185673276807403862670020...
MATHEMATICA
kappa[t_] := -1 - 1/Pi*Arg[ RiemannSiegelZ'[t] - I*RiemannSiegelZ[t]*RiemannSiegelTheta'[t]]; digits = 60; t0[n_] := t0[n] = (t /. FindMinimum[kappa[t], {t, 1}, WorkingPrecision -> n] [[2]]) // RealDigits[#, 10, digits] & // First; t0[digits]; t0[n = 2*digits]; While[t0[n] != t0[n - digits], n = n + digits]; t0[n]
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, May 22 2013
STATUS
approved