# Integer matrices related to Liouville's function

Czechoslovak Mathematical Journal (2013)

- Volume: 63, Issue: 1, page 39-46
- ISSN: 0011-4642

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topOon, Shea-Ming. "Integer matrices related to Liouville's function." Czechoslovak Mathematical Journal 63.1 (2013): 39-46. <http://eudml.org/doc/252544>.

@article{Oon2013,

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.},

author = {Oon, Shea-Ming},

journal = {Czechoslovak Mathematical Journal},

keywords = {Liouville's function; determinant; LU decomposition; Liouville's function; determinant; LU decomposition},

language = {eng},

number = {1},

pages = {39-46},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Integer matrices related to Liouville's function},

url = {http://eudml.org/doc/252544},

volume = {63},

year = {2013},

}

TY - JOUR

AU - Oon, Shea-Ming

TI - Integer matrices related to Liouville's function

JO - Czechoslovak Mathematical Journal

PY - 2013

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 63

IS - 1

SP - 39

EP - 46

AB - 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.

LA - eng

KW - Liouville's function; determinant; LU decomposition; Liouville's function; determinant; LU decomposition

UR - http://eudml.org/doc/252544

ER -

## References

top- Apostol, T. M., Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics New York-Heidelberg-Berlin: Springer (1976). (1976) Zbl0335.10001MR0434929
- 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. (2009) Zbl1190.15024MR2551085
- Higham, N. J., 10.1137/1029112, SIAM Rev. 29 575-596 (1987). (1987) Zbl0635.65049MR0917696DOI10.1137/1029112
- Hong, Y. P., Pan, C.-T., A lower bound for the smallest singular value, Linear Algebra Appl. 172 27-32 (1992). (1992) Zbl0768.15012MR1168494
- Landau, E., Handbuch der Lehre von der Verteilung der Primzahlen. Erster Band, Leipzig u. Berlin: B. G. Teubner. X (1909). (1909)
- Redheffer, R., Eine explizit lösbare Optimierungsaufgabe, Numer. Meth. Optim.-Aufg. 36 213-216 (1977). (1977) Zbl0363.65062MR0468170
- Tenenbaum, G., Introduction à la Théorie Analytique et Probabiliste des Nombres, Cours Spécialisés 1 Paris: Société Mathématique de France (1995). (1995) Zbl0880.11001MR1366197

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