Some series representing the Riemann zeta function

Abstract

Series, converging geometrically fast on the complex plane toward the Riemann zeta function, are provided. They are expressed using a sequence, depending on the complex variable and on an integer at least 2, and obeying a linear recurrence. The asymptotic to all orders in inverse powers of the indexing integer is obtained. Related sequences were first encountered in theoretical computer science decades ago. The tools to obtain their asymptotic expansion to all orders were developed by the author in a special case devoted to the Euler-Mascheroni constant. These tools are applied here in a more general context.