Czechoslovak Mathematical Journal, Vol. 63, No. 1, pp. 39-46, 2013

Integer matrices related to Liouville's function

Shea-Ming Oon

Shea-Ming Oon, Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia, e-mail: oonsm@um.edu.my

Abstract: In this note, we construct some integer matrices with determinant equal to certain summation form of Liouville's function. Hence, it offers a possible alternative way to explore the Prime Number Theorem by means of inequalities related to matrices, provided a better estimate on the relation between the determinant of a matrix and other information such as its eigenvalues is known. Besides, we also provide some comparisons on the estimate of the lower bound of the smallest singular value. Such discussion may be extended to that of Riemann hypothesis.

Keywords: Liouville's function, determinant, LU decomposition

Classification (MSC 2010): 11A25, 11C20, 15A15, 15B36


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