## Abstract

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 ^{2}N (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(log^{2}_{2}N/log N)], where ζ_{∞} = Π_{k ≥ 2}ζ(k) ∼ 2.3, and in τ(N) = O(log_{2}N) the multiplicative structure of N is recovered.

Original language | English |
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Pages (from-to) | 153-198 |

Number of pages | 46 |

Journal | Linear Algebra and Its Applications |

Volume | 81 |

Issue number | C |

DOIs | |

State | Published - Sep 1986 |