Geodesic
Integer matrices related to Liouville's function
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 1, pp. 39-46 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.1007/s10587-013-0002-8
Classification : 11A25, 11C20, 15A15, 15B36
Keywords: Liouville's function; determinant; LU decomposition 
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Oon, Shea-Ming. Integer matrices related to Liouville's function. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 1, pp. 39-46. doi: 10.1007/s10587-013-0002-8

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