Prescribed Eigenvalues in Power Bands of Finite Prime-Cover Graphs
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)…
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