  Recently, we have established the generalized Li's criterion equivalent to
the Riemann hypothesis, viz. demonstrated that the sums over all non-trivial
Riemann function zeroes k_n,b=Sum_rho(1-(1-((rho+b)/(rho-b-1))**n) for any real
b not equal to -1/2 are non-negative if and only if the Riemann hypothesis
holds true; arXiv:1304.7895 (2013); Ukrainian Math. J., 66, 371 - 383, 2014.
(Famous Li's criterion corresponds to the case b=0 (or b=1) here). This makes
timely the detailed studies of these sums, and in particular also the study of
their asymptotic for large n. This question, assuming the truth of RH, is
answered in the present Note. We show that on RH, for large enough n, for any
real b not equal to -1/2, one has:
k_n,b=Sum_rho(1-(1-((rho+b)/(rho-b-1))**n)=0.5*abs(2b+1)*n*ln(n)+0.5*abs(2b+1)*(gamma-1-ln(2*pi/abs(2b+1))*n+o(n),
where gamma is Euler-Mascheroni constant.
