Complements to Liʹs Criterion for the Riemann Hypothesis Original Research Article

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 Liʹs 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 Liʹs paper, via the Guinand–Weil explicit formula, and relate the conjectural positivity ofλnto Weilʹs criterion for the Riemann Hypothesis.