Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 182-02. A Power Saving for Polynomial Differences at Prime Arguments
Manuscript map · lines 5725–5734
OpenAI@2026-10-07#rem:oai-p-182-02
Authors: OpenAI. Family 182.
Title: A Power Saving for Polynomial Differences at Prime Arguments.
Abstract: Let h be a fixed integer polynomial of degree at least two with positive leading coefficient, having a unit root modulo every positive integer. We prove that any set $A{}subseteq{}{1,{}ldots,N{}}$ whose differences avoid all nonzero values $h(p)$ at primes satisfies $|A|{}le C_hN{}{1-c_h}$, where $c_h{}gt 0$ and $C_h{}ge1$ depend only on h. Thus the local unit-root condition gives a fixed power saving even when polynomial arguments are restricted to primes. The proof uses the companion zero-free half-plane theorem for Dirichlet L-functions to obtain the required prime-distribution estimates.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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