Entire functions with undecidable arithmetic properties

Abstract

A basic problem in transcendental number theory is to determine the arithmetic properties of analytic functions of the form f(z)=∑k=0∞akzk where the coefficients ak∈K belong to an algebraic number field. In particular, one of the most basic problems is to determine if f(α) is algebraic or transcendental for non-zero algebraic arguments α. For example, if f(z) is a transcendental Mahler function, then under generic conditions f(α) is transcendental for all non-zero algebraic numbers with |α|<1. Also, if f(z) is an E-function, then there exist algorithms which completely determine the arithmetic properties of f(n)(α) for non-zero algebraic numbers α. In contrast to these and other algorithmic results, we construct three functions f(z), g(z), and h(z) with computable rational coefficients for which no algorithms exist that determine if f(n)∈Q, g(n)(1)∈Q, or ∫01h(z)zndz∈Q for integral n≥0. Our results are an application of an undecidable variant of the Collatz Problem due to Lehtonen [9].