Asymptotics of Keiper-Li coefficients

Abstract

We show that the Riemann Hypothesis is equivalent to the assertion $(y_m)\in\ell_2$ where $y_m$ is defined by \[ \lambda_m=\frac12(\log m+\gamma-\log(2\pi)-1)+y_m, \] and $m\lambda_m$ represents the numbers in Xian-Jin Li's criterion. This confirms and further sharpens a conjecture of J. B. Keiper. We also present some other hypotheses equivalent to the Riemann Hypothesis.