Logarithmic-exponential power series. (English) Zbl 0924.12007

van den Dries, Lou; Macintyre, Angus; Marker, David

J. Lond. Math. Soc., II. Ser. 56, No. 3, 417-434 (1997).

This paper builts on the earlier one: “The elementary theory of restricted analytic fields with exponentiation” [Ann. Math. (2) Ser. 140, No. 1, 183-205 (1994; Zbl 0837.12006)] written by the same authors. In the earlier paper the theory \(T_{\text{an,exp}}\) of the structure \(\mathbb{R}_{\text{an,exp}}\) of the field of real numbers endowed with the restriction to \([-1,+1]^m\) for all \(m\)-ary real valued functions \(f\), analytic in a neighbourhood of \([-1,+1]^m\), together with the exponentiation exp was studied. In this paper the authors show that generalized power series can be used to give an algebraic construction of nonstandard models of \(T_{\text{an,exp}}\). They use these models in order to show that the compositional inverse to \(x\mapsto (\log x)(\log\log x)\) is not asymptotic to a composition of semi-algebraic functions, log and exp. This answers a question of Hardy. They also show that certain functions on \((0,+\infty)\), including \(\Gamma(x)\), \(\int_0^x e^{t^2} dt\), and \(\int_0^\infty e^{-t} (t+x)^{-1}dt\) are not definable in \(\mathbb{R}_{\text{an,exp}}\). By a different method they show that the Riemann zeta function restricted to \((1,+\infty)\) is not definable in \(\mathbb{R}_{\text{an,exp}}\).

Reviewer: A.Prestel (Konstanz)

MSC:

12L12	Model theory of fields
03C60	Model-theoretic algebra
03H05	Nonstandard models in mathematics
12J15	Ordered fields

Keywords:

Hardy fields; exponentiation; generalized power series; nonstandard models; Riemann zeta-function

Citations:

Zbl 0837.12006