From asymptotic to closed forms for the Keiper/Li approach to the Riemann Hypothesis

André Voros (1)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681

The Riemann Hypothesis (RH)-that all nonreal zeros of Riemann's zeta function shall have real part 1/2-remains a major open problem. Its most concrete equivalent is that an infinite sequence of real numbers, the Keiper-Li constants, shall be everywhere positive (Li's criterion). But those numbers are analytically elusive and strenuous to compute, hence we seek simpler variants. The essential sensitivity to RH of that sequence lies in its asymptotic tail; then, retaining this feature, we can modify the Keiper-Li scheme to obtain a new sequence in elementary closed form. This makes for a more explicit analysis, with easier and faster computations. We can moreover show how the new sequence will signal RH-violating zeros if any, by observing its analogs for the Davenport-Heilbronn counterexamples to RH.

Domaines: Physique [physics]

André Voros. From asymptotic to closed forms for the Keiper/Li approach to the Riemann Hypothesis. 2022. ⟨cea-03673957⟩