1. Entire functions with undecidable arithmetic properties.
- Author
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Ferguson, Timothy
- Subjects
- *
ALGEBRAIC numbers , *IRRATIONAL numbers , *TRANSCENDENTAL functions , *NUMBER theory , *ARITHMETIC , *ARITHMETIC functions , *ALGEBRAIC fields - Abstract
A basic problem in transcendental number theory is to determine the arithmetic properties of analytic functions of the form f (z) = ∑ k = 0 ∞ a k z k where the coefficients a k ∈ 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 ∫ 0 1 h (z) z n d z ∈ Q for integral n ≥ 0. Our results are an application of an undecidable variant of the Collatz Problem due to Lehtonen [9]. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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