Asymptotic of the generalized Li's sums which non-negativity is equivalent to the Riemann Hypothesis

Abstract

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.