Complements to Li's criterion for the Riemann hypothesis

Abstract

In a recent paper Xian-Jin Li showed that the Riemann Hypothesis holds if and only if λn = Σp [ 1 - (1 - 1/p)n] has λn > 0 for n = 1, 2, 3, ... where p 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 p and therefore is not specific to zeta functions. We also give an arithmetic formula for the numbers λn in Li's paper, via the Guinand-Weil explicit formula, and relate the conjectural positivity of λn to Weil's criterion for the Riemann Hypothesis. © 1999 Academic Press.