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 ifλn=∑ρ[1−(1−1/ρ)n] hasλn>0 forn=1, 2, 3, … whereρruns 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 numbersρand therefore is not specific to zeta functions. We also give an arithmetic formula for the numbersλnin Li's paper, via the Guinand–Weil explicit formula, and relate the conjectural positivity ofλnto Weil's criterion for the Riemann Hypothesis.