Submission date: 19 February 2024

Abstract:

Given an o-minimal expansion ℝ_𝒜 of the real ordered field, generated by a generalized quasianalytic class 𝒜, we construct an explicit truncation closed ordered differential field embedding of the Hardy field of the expansion ℝ_{𝒜,exp} of ℝ_𝒜 by the unrestricted exponential function, into the field 𝕋 of transseries. We use this to prove some non-definability results. In particular, we show that the restriction to the positive half-line of Euler's Gamma function is not definable in the structure ℝ_{an^*,exp}, generated by all convergent generalized power series and the exponential function, thus establishing the non-interdefinability of the restrictions to a neighbourhood of + ∞ of Euler's Gamma and of the Riemann Zeta function.

Mathematics Subject Classification: 03C64, 26E10, 03C10, 12J15

Keywords and phrases:

Full text arXiv 2402.12073: pdf, ps.