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Integer matrices related to Liouville’s function. (English) Zbl 1274.11012

Let \(n\) be a positive integer and let \(\Omega (n)\) be the number of prime factors of \(n\). The Liouville function \(\lambda \) is defined by \(\lambda (n)=(-1)^{\Omega (n)}\). In the paper under review, the authors define two matrices and show that their determinants are related to \(\lambda (n)\). For example, define \[ L(n)\:=\sum _{k=1}^{n}\lambda (k), \] and an \(n\times n\) matrix \(S_n=(s_{ij})\) by \[ s_{ij}=\begin{cases} 1 & \text{ if }j=1 \text{ or } j/i \text{ is a square-free integer},\\ 0 & \text{ else.} \end{cases} \] Then the authors show that \(\det \left (S_n\right)=L(n)\). They also give a discussion of the relationship of these theorems to the Prime Number Theorem. The proofs are elementary.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11C20 Matrices, determinants in number theory
15A15 Determinants, permanents, traces, other special matrix functions
15B36 Matrices of integers
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References:

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