Entire functions with undecidable arithmetic properties

A basic problem in transcendental number theory is to determine the arithmetic properties of analytic functions of the form f(z) = Sigma(k=0) a(k)z(k) where the coefficients a(k) is an element of K belong to an algebraic number field. In particular, one of the most basic problems is to determine if f (alpha) is algebraic or transcendental for non-zero algebraic arguments alpha. For example, if f(z) is a transcendental Mahler function, then under generic conditions f (alpha) is transcendental for all non-zero algebraic numbers with vertical bar alpha vertical bar< 1. Also, if f (z) is an E-function, then there exist algorithms which completely determine the arithmetic properties of f((n))(alpha) for non-zero algebraic numbers alpha. In contrast to these and other algorithmic results, we construct three functions f (x), g(z), and h(z) with computable rational coefficients for which no algorithms exist that determine if f (n) is an element of Q, g((n)) is an element of Q, or integral(1)(0) h(z) z(n) dz is an element of Q for integral n >= 0. Our results are an application of an undecidable variant of the Collatz Problem due to Lehtonen [9]. (C) 2020 Elsevier Inc. All rights reserved.