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
-0.6702597987685995028839164119686674474803927900974349173...
MATHEMATICA
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]
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, May 22 2013
STATUS
approved