On the optimal constant in the Montgomery-Vaughan weighted Hilbert inequality
Abstract
We give a proof that the optimal constant in the weighted Hilbert inequality is strictly greater than $π$, answering in the negative a question asked by Montgomery and Vaughan. The proof proceeds via an explicit limiting counterexample.
Disclosure
“pitality. Use of large language models. ChatGPT 5.6 Sol has been used in this proof. The numerical experiments described in Remark 2.3 formed the start of this project. These began in 2024 but it was only recently that programs written by ChatGPT 5.6 Sol uncovered unexpected high-dimensional behavior. A further back and forth about these numerics led ultimately to the proof of Theorem 1.1 given in Section 2. The proof that appears here is a modification and simplification of a proo”
PDF page 2
- Classification
- Substantial proof generation
- Multiplier
- 10
- Verified
Structural counts
Pages 6 pdf
Theorems 1 source
Lemmas 0 source
Propositions 0 source
Corollaries 0 source
Definitions 0 source
Displayed equations 25 source
Bibliography entries 12 source
Appendix pages 0 estimated
Count notes
- Source counts use the expanded primary TeX file weightedHilbert_arxiv.Aug12.tex.
- Appendix pages include the first PDF page with an explicit Appendix heading through the final page.