## 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

### Keywords:

Liouville’s function; determinant; LU decomposition
Full Text:

### References:

 [1] T. M. Apostol: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics, New York-Heidelberg-Berlin: Springer, 1976. [2] O. Bordellès, B. Cloître: A matrix inequality for Möbius functions. JIPAM, J. Inequal. Pure Appl. Math. 10 (2009), Paper No. 62, pp. 9, electronic only. · Zbl 1190.15024 [3] N. J. Higham: A survey of condition number estimation for triangular matrices. SIAM Rev. 29 (1987), 575–596. · Zbl 0635.65049 [4] Y. P. Hong, C.-T. Pan: A lower bound for the smallest singular value. Linear Algebra Appl. 172 (1992), 27–32. · Zbl 0768.15012 [5] E. Landau: Handbuch der Lehre von der Verteilung der Primzahlen. Erster Band. Leipzig u. Berlin: B. G. Teubner. X, 1909. · JFM 40.0232.08 [6] R. Redheffer: Eine explizit lösbare Optimierungsaufgabe. Numer. Meth. Optim.-Aufg. 36 (1977), 213–216. · Zbl 0363.65062 [7] G. Tenenbaum: Introduction à la Théorie Analytique et Probabiliste des Nombres. Cours Spécialisés 1, Paris: Société Mathématique de France, 1995.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.