The Keiper-Li Criterion for the Riemann Hypothesis and Generalized Lambert Functions

Abstract

Keiper [1] and Li [2] published independent investigations of the connection between the Riemann hypothesis and the properties of sums over powers of zeros of the Riemann zeta function. Here we consider a reframing of the criterion, to work with higher-order derivatives ξ r of the symmetrized function ξ ( s ) at s = 1/2, with all ξ r known to be positive. The reframed criterion requires knowledge of the asymptotic properties of two terms, one being an infinite sum over the ξ r . This is studied using known asymptotic expansions for the ξ r , which give the location of the summand as a relationship between two parameters. This relationship needs to be inverted, which we show can be done exactly using a generalized Lambert function. The result enables an accurate asymptotic expression for the value of the infinite sum.