Multisummability for generalized power series

Abstract

We develop multisummability, in the positive real direction, for generalized power series with natural support, and we prove o-minimality of the expansion of the real field by all multisums of these series. This resulting structure expands both $\mathbb{R}_{\mathcal{G}}$ and the reduct of $\mathbb{R}_{\mathrm{an}^*}$ generated by all convergent generalized power series with natural support; in particular, its expansion by the exponential function defines both the Gamma function on $(0,\infty)$ and the Zeta function on $(1,\infty)$.