Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Liouville's function; determinant; LU decomposition
Liouville's function; determinant; LU decomposition
Summary:
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.
References:
[1] Apostol, T. M.: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics New York-Heidelberg-Berlin: Springer (1976). MR 0434929 | Zbl 0335.10001
[2] Bordellès, O., Cloître, B.: A matrix inequality for Möbius functions. JIPAM, J. Inequal. Pure Appl. Math. 10 (2009), Paper No. 62, pp. 9, electronic only. MR 2551085 | Zbl 1190.15024
[3] Higham, N. J.: A survey of condition number estimation for triangular matrices. SIAM Rev. 29 575-596 (1987). DOI 10.1137/1029112 | MR 0917696 | Zbl 0635.65049
[4] Hong, Y. P., Pan, C.-T.: A lower bound for the smallest singular value. Linear Algebra Appl. 172 27-32 (1992). MR 1168494 | Zbl 0768.15012
[5] Landau, E.: Handbuch der Lehre von der Verteilung der Primzahlen. Erster Band. Leipzig u. Berlin: B. G. Teubner. X (1909).
[6] Redheffer, R.: Eine explizit lösbare Optimierungsaufgabe. Numer. Meth. Optim.-Aufg. 36 213-216 (1977). MR 0468170 | Zbl 0363.65062
[7] Tenenbaum, G.: Introduction à la Théorie Analytique et Probabiliste des Nombres. Cours Spécialisés 1 Paris: Société Mathématique de France (1995). MR 1366197 | Zbl 0880.11001
Search
Partner of
