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 ξr of the symmetrized function ξ(s) at s = 1/2, with all ξr known to be positive. The reframed criterion requires knowledge of the asymptotic properties of two terms, one being an infinite sum over the ξr. This is studied using known asymptotic expansions for the ξr, 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.