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Logarithmic-exponential power series. (English) Zbl 0924.12007
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 $$\mathbb{R}_{\text{an,exp}}$$.

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