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 n = P 1?(1?1==) n has n > 0 for n = 1; 2; 3; : : : where 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 and therefore is not speciic 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. 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 ? 1) ?s=2 ? s 2 (s) and the sequence n = 1 (n ? 1)! d n ds n s n?1 log (s) s=1 for n = 1; 2; 3; : : : , in the form that the Riemann Hypothesis is equivalent to the statement that n > 0 for every positive integer n. He also showed that an identical result applies to the Riemann Hypothesis for the Dedekind zeta function of a number eld. The number n can be written in terms of the complex zeros of the Riemann zeta function (or Dedekind zeta function) as n = X 1 ? 1 ? 1 n where the sum* over is understood as X = lim T!1 X j=()jjT : * We denote by <(z) and =(z) the real and imaginary part of the complex number z