Precision loss of zeta(2)-zetahurwitz(2,N+1) in some range of N ?

[Gottfried Helms <helms@uni-kassel.de>]

I recently came across this, when I wanted to
use zeta and zetahurwitz for sum of reciprocal squares
up to large N. My default precision is always 200 dec
digits and so this gave a signal... (I never
observed such a problem of loss of precision with the
harmonic numbers and the psi()-function)


H2o(m)=sum(k=1,m,1.0/k^2)
H2(m)=zeta(2)-zetahurwitz(2,m+1)

H2o(20)-H2(20)
H2o(200)-H2(200)
H2o(2000)-H2(2000)
H2o(20000)-H2(20000)
H2o(200000)-H2(200000)

%1412 = -1.66345512051 E-211
%1414 =  2.20232112736 E-141
%1416 =  5.49367255720 E-177
%1418 =  5.57935952213 E-202
%1420 = -2.37743382187 E-207

Gottfried Helms