Abstract

In a recent paper Xian-Jin Li showed that the Riemann Hypothesis holds if and only if n = P ae \Theta 1 \Gamma (1 \Gamma 1=ae) n has n ? 0 for n = 1; 2; 3; : : : where ae 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 ae and therefore is not specific to zeta functions. We also give an arithmetic formula for the numbers n in Li's paper, via the Guinand-Weil explicit formula and relate the conjectural positivity of n to Weil's criterion for the Riemann Hypothesis. 1990 Mathematics Subject Classification 11M26, 11R42. x1. Introduction. In a recent paper Xian-Jin Li [3] obtained an interesting criterion for the validity of the Riemann Hypothesis. His criterion can be stated in terms of the Riemann ¸ -function ¸(s) = 1 2 s(s \Gamma 1)ß \Gammas=2 \Gamma i s 2 j i(s) and the sequence n = 1 (n \Gamma 1)! d n ds n \Theta s n\Gam...

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