Integer Coefficient Power Series with Prescribed Zero Sets

Abstract

We prove that a discrete effective divisor on the open unit disk $\mathbb{D}$ is the zero divisor of a holomorphic function on $\mathbb{D}$ with integer Taylor coefficients if and only if it is invariant under complex conjugation. The construction uses a one-parameter deformation of the Weierstrass elementary factors in which each modified factor of order $n$ leaves all Taylor coefficients of degree $\leq n$ unchanged while shifting the coefficient of degree $n+1$ by a controlled affine amount. These modified factors act as elementary jet-correction operators: the triangular structure of the coefficient map permits an inductive rounding scheme compatible with canonical-product convergence. As a consequence, every holomorphic function on $\mathbb{D}$ differs from one with Gaussian-integer Taylor coefficients by multiplication by a nowhere-vanishing holomorphic factor.

Disclosure

“associated ring-theoretic consequences, remain to be worked out. Tool and Computational Resource Disclosure This manuscript was drafted and revised with the assistance of Claude (Anthropic; primarily Claude Opus 4.6), a large language model used interactively through Claude Code. Claude contributed to composing and revising portions of the exposi- tion, and produced the accompanying Lean 4 formalization of Theorems 1 and 2 referenced in the footnote on the first page, built u”

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2026-08-24T00:34:09.244Z

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