HEXAGONbeta

Spectral synthesis for the Báez–Duarte step space

Primarily AI-generated textHuman understanding: all partsmath.NT — Number Theory

Contributed by Will Howe, Grant Huffman ↗, Jensen Kohlmeyer ↗, Liam Kruer ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

We prove that the closed Báez–Duarte subspace of the Mellin step space is exactly the common kernel of the derivative evaluations prescribed by the zeros of the Riemann zeta function in Re⁡s>1/2\operatorname{Re}s>1/2, with their multiplicities. No finiteness or sparsity assumption on this zero set is required. The proof combines a uniform fractional regularity estimate for Mellin transforms of step functions with a comparison obtained by reflecting zeros in a shifted vertical line. These estimates produce a holomorphic Hardy-space family whose membership in the Báez–Duarte subspace continues from large positive shifts to every sufficiently small positive shift. A uniformly bounded weak limit then gives the synthesis theorem.

Provenance statement

The manuscript was primarily AI-generated using OpenAI tools, which assisted in refining the argument, checking intermediate steps, and preparing the mathematical exposition. All parts have been checked and understood by a human.

References

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  5. span_of_dilates , The Báez-Duarte subspace has codimension equal to the number of zeta zeros off the critical line, with distance and rate bounds not assuming RH , rStation, Paper 1, version 12, October 27, 2026. https://rstation.ai/problems/riemann_hypothesis/papers/1.link
  6. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function , 2nd ed., revised by D. R. Heath-Brown, Clarendon Press, Oxford, 1986.

Version history

  1. v1 · 9 pagesSubmitted by Jensen KohlmeyerInitial depositCurrentNov 10, 2026