Sharp Circular Sampling and Derivative Period Polynomials

Abstract

We determine the exact maximal reflected zero region that forces centered binomial samples of a balanced entire function to have all zeros on the unit circle. In degree $d\ge2$, this region is \[ Ω_d=\left\{a+ib:\ a^2-\frac{b^2}{d-1}\le\frac d4\right\}. \] The finite theorem is sharp already for a single reflected zero pair, and a phase-preserving canonical-product approximation extends it to balanced entire functions of order at most one. De Bruijn strip contraction and projective Hermite--Kakeya--Obreschkoff theory then give the exact common-zero obstruction, simplicity, strict interlacing of consecutive derivative samples, and a monotone real-pencil root flow. As an application, we prove the derivative-period-polynomial unit-circle theorem for completed $L$-functions of primitive holomorphic newforms, in every derivative order and for arbitrary level and nebentypus. After the standard normalization, every zero of \[ \sum_{j=0}^{k-2}\binom{k-2}{j}Λ^{(m)}(f,j+1)z^j \] lies on the unit circle for every weight $k\ge4$, level, nebentypus, and derivative order $m\ge0$. In particular, this proves the full-polynomial unit-circle conjecture of Diamantis and Rolen in its original level-one setting and extends it to arbitrary level and nebentypus. The same source-side theorem also gives simplicity, strict interlacing, and, for each fixed derivative order, conductor-uniform quantitative localization in the weight aspect.

Disclosure

“ypus for every fixed derivative order and to retain enough phase information to compare consecutive orders. Use of AI-assisted tools For transparency, during the preparation of this manuscript the author used OpenAI ChatGPT to discuss possible gaps in arguments, improve exposition, and edit LaTeX and English prose. The tool was not used as a source of mathematical results or as a formal proof verifier. All proofs, computations, statements, and references were”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-24T00:34:09.244Z

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