Prescribed Eigenvalues in Power Bands of Finite Prime-Cover Graphs
A mix of human-written and AI-generated textHuman understanding: all partsmath.CO — Combinatoricsmath.NT — Number Theory
Contributed by Idris Ali Shaik ↗
Version 1 / Oct 20, 2026 / CC BY 4.0
Abstract
Let G_X be the graph on the positive integers at most X, with an edge between n and np whenever p is prime and np <= X. Write lambda_1(X) >= lambda_2(X) >= ... > 0 for its positive adjacency eigenvalues, counted with multiplicity. We prove that, for every fixed 21/61 < theta < 1/2 and every integer X^theta/log X <= a <= X^theta,
lambda_a(X)^2 = pi(X/a) + O_theta(a/log X),
where pi is the prime-counting function. This identifies the eigenvalue at the prescribed rank a throughout a polynomially growing band of centres. For every 0 < theta < 1/2, ordered mean-square bounds give a weaker estimate outside o(X^theta) centres below X^theta. When 1/3 < theta < 1/2, the same O_theta(a/log X) error holds outside o(X^theta) centres using only global prime estimates. The results extend to deletion of any fixed finite set of prime directions.
Edges labelled by primes exceeding sqrt(X) form disjoint stars. Retaining their first-exit interactions gives local comparison operators and a family of approximate eigenvectors. Bounds on their overlaps and residuals control eigenvalue counts across clusters, allowing ranks to be recovered where the global perturbation bound no longer separates neighbouring star levels. The Baker--Harman--Pintz lower bound and the Brun--Titchmarsh upper bound supply the short-interval prime counts used in the uniform every-centre theorem.
Provenance statement
During the preparation of this work, the author used OpenAI Codex, including AI-assisted research and coding tools, to support mathematical exploration, proof-development assistance, manuscript revision, finite diagnostic design, and Lean 4 formalization. The author reviewed and edited the output as needed and takes full responsibility for the content of this manuscript.
Formalizations
Tools used
- OpenAI
- Codex
- Lean
- Lean
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Version history
- v1Submitted by Idris Ali ShaikInitial depositCurrentOct 18, 2026