Riemann's hypothesis as an eigenvalue problem

The matrix A_N=(a_m,n)_2≤m,n≤N, where a_m,n=m - 1 if m n and a_m,n = -1 if m {does not divide} n, has the determinant N!Σ_{1 ≤ n ≤ N} μ(n) n, μ the Möbius function, and thus is closely connected with the Riemann hypothesis, which is true if and only if det A_N = O(N!N ^{-1 2 + ε}) for every ε > 0 (Theorem 1). det A_N is the product of the eigenvalues λ_n (1 ≤ n ≤ N - 1) of A_N; therefore it might be of interest to find properties of these eigenvalues. A theorem of Gerschgorin shows immediately |λ_n| ≤ N - 1/N for 1 ≤ n ≤ N - 1 (Theorem 2), and a discussion of the characteristic polynomial χ_N(x) of A_N gives n ≤ λ_n < n + 1 for all n with at most N - 2 N exceptions. In particular, all natural numbers n such that N/3 < n ≤ N/2 turn out to be eigenvalues of A_N (Theorem 5). The power sums of the eigenvalues have the property ∑ 1≥n≥N-1λ^k_n ≥1^k+2^k+⋯+(N-1)^k for all k≥0, ≥2^k+3^k+⋯+N^k for k≥N 1 2 and cN log ²N (Theorem 9), so that g^*_{k, N}, defined by ∑ n=1 N-1λ^k_n = ∑ n=2 N-1n^k + g^*_{k, N}N^k, is-with regard to n ≤ λ_n < n + 1-a measure for the uniformity of the distribution of the eigenvalues λ_n in comparison with their indices n. A Tauberian theorem shows that the values g^*_{k, N} almost approach 1 (Corollary of Theorem 8), but lim_{k → ∞}g^*_{k, N} = 0. If the Riemann hypothesis is true, Theorem 1 might suggest λ_n ∼ n + 1 2, but formula (2) in the Corollary of Theorem 11 hints at a slight average shift of the λ_n's to n + 1. This is exemplified by the largest eigenvalue λ_max of A_N (Theorem 11): λ_max = N - [τ(N)/ζ_∞log N][1 + O(log²₂N/log N)], where ζ_∞ = Π_{k ≥ 2}ζ(k) ∼ 2.3, and in τ(N) = O(log₂N) the multiplicative structure of N is recovered.