Yes, I mean the coefficients of a power series about 0. I don't understand the comment about diverging at 1. But fair enough, asking for a specific question. Specific question: ------------------ Suppose there is an infinite power series f(z) = Sum_{0<=n<oo} c_n z^n that has integer coefficients {c_n in Z| n >= 0} with infinitely many c_n nonzero, and that converges for |z| < R, some R > 0. Let r be a non-algebraic zero of analytic function f(z) in the interior of its region of convergence: ----- f(r) = 0, |r| < R but P(r) != 0 for all nontrivial integer polynomials P(z). ----- SPECIFIC QUESTION: Does there exist a non-algebraic number in C that is *not* such a zero of an infinite power series with integer coefficients? —Dan Andy Latto wrote: ----- What do you mean by the "coefficients" of an analytic function? The coefficients of its power series centered at the origin? Such a series, unless it has only finitely many terms (in which case its roots are algebraic0, will diverge at 1, so one ting you can say about the set is "it's contained in the unit circle" I'm not sure what you mean by "what *is* the set of...". I don't think this set will have any simpler or more familiar description. It might be more fruitful to ask about some particular properties of the set. For example, I'm pretty sure it's uncountable, but don't have an immediate proof. On Sat, Feb 10, 2018 at 4:16 PM, Dan Asimov <dasimov@earthlink.net> wrote:

An algebraic number of degree d is a root of some degree d integer polynomial that is irreducible (not the product of two polynomials of positive degree).

Question 1: ----------- What is the set of non-algebraic roots of analytic functions with integer coefficients?