Riemann's hypothesis as an eigenvalue problem

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Abstract

The matrix AN=(am,n)2≤m,n≤N, where am,n=m - 1 if m n and am,n = -1 if m {does not divide} n, has the determinant N!Σ1 ≤ 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 AN = O(N!N -1 2 + ε) for every ε > 0 (Theorem 1). det AN is the product of the eigenvalues λn (1 ≤ n ≤ N - 1) of AN; therefore it might be of interest to find properties of these eigenvalues. A theorem of Gerschgorin shows immediately |λn| ≤ N - 1/N for 1 ≤ n ≤ N - 1 (Theorem 2), and a discussion of the characteristic polynomial χN(x) of AN gives n ≤ λn < 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 AN (Theorem 5). The power sums of the eigenvalues have the property ∑ 1≥n≥N-1λkn ≥1k+2k+⋯+(N-1)k for all k≥0, ≥2k+3k+⋯+Nk for k≥N 1 2 and cN log 2N (Theorem 9), so that g*k, N, defined by ∑ n=1 N-1λkn = ∑ n=2 N-1nk + g*k, NNk, is-with regard to n ≤ λn < n + 1-a measure for the uniformity of the distribution of the eigenvalues λn in comparison with their indices n. A Tauberian theorem shows that the values g*k, N almost approach 1 (Corollary of Theorem 8), but limk → ∞g*k, N = 0. If the Riemann hypothesis is true, Theorem 1 might suggest λn ∼ n + 1 2, but formula (2) in the Corollary of Theorem 11 hints at a slight average shift of the λn's to n + 1. This is exemplified by the largest eigenvalue λmax of AN (Theorem 11): λmax = N - [τ(N)/ζlog N][1 + O(log22N/log N)], where ζ = Πk ≥ 2ζ(k) ∼ 2.3, and in τ(N) = O(log2N) the multiplicative structure of N is recovered.

Original languageEnglish
Pages (from-to)153-198
Number of pages46
JournalLinear Algebra and Its Applications
Volume81
Issue numberC
DOIs
StatePublished - Sep 1986

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