Some geometric series for Euler's constant

Abstract

We provide representations of Euler's constant $\gamma =\mathrm{0.577...}$ as series which converge geometrically fast (but use a certain sequence whose computation induces a quadratic cost). The asymptotic oscillations of these coefficients are determined to all orders. A result of independent interest, about sufficient conditions for the validity, in the case of unbounded parameters, for the Tricomi-Erdélyi asymptotic expansion of the ratio of two Gamma functions, is established for that purpose.

32 pages, 2 figures. v2 incorporated advances from arXiv:2602.19727 and arXiv:2604.24754, v3 and v4 addressed the fact that the e_m's appeared decades ago in theoretical computer science and v5 provided their asymptotics to all orders; arXiv:2602.05511 has now been rewritten to use the tools from v5. v6 simply updates references