{"id":"hexagon/2610.00358","pdf":"https://hexagonmath.org/pdf/2610.00358v1","url":"https://hexagonmath.org/abs/2610.00358v1","tags":["math.NT","msc:05C50","msc:11N05","matrix perturbation","ordered eigenvalues","prime number theorem","prime-cover graphs","short intervals","spectral graph theory"],"score":0,"title":"Prescribed Eigenvalues in Power Bands of Finite Prime-Cover Graphs","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Idris Ali Shaik"],"reviews":0,"subject":"math.CO","summary":"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 = 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)…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Codex"],"discussed":0,"published":"2026-10-20","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":true,"formalizations":["https://palomar-registry.org/entry?id=PALOMAR-2026-09-10-000004&version=1"],"human_understanding":"all"}