Complements to Li's criterion for the Riemann hypothesis

In a recent paper Xian-Jin Li showed that the Riemann Hypothesis holds if and only if lambda(n) = Sigma(rho){[}1 - (1 - 1/rho)(n)] has lambda(n) > 0 for n = 1,2, 3, ... where rho runs over the complex zeros of the Riemann zeta function. We show that Li's criterion follows as a consequence of a general set of inequalities for an arbitrary multiset of complex numbers rho and therefore is not specific to zeta functions. We also give an arithmetic formula for the numbers lambda(n) in Li's paper, via the Guinand-Weil explicit formula, and relate the conjectural positivity of lambda(n) to Well's criterion for the Riemann Hypothesis. (C) 1999 Academic Press.