| Title:
|
Integer matrices related to Liouville's function (English) |
| Author:
|
Oon, Shea-Ming |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
1 |
| Year:
|
2013 |
| Pages:
|
39-46 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Liouville's function |
| Keyword:
|
determinant |
| Keyword:
|
LU decomposition |
| MSC:
|
11A25 |
| MSC:
|
11C20 |
| MSC:
|
15A15 |
| MSC:
|
15B36 |
| idZBL:
|
Zbl 1274.11012 |
| idMR:
|
MR3035495 |
| DOI:
|
10.1007/s10587-013-0002-8 |
| . |
| Date available:
|
2013-03-01T16:00:47Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143168 |
| . |
| Reference:
|
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| Reference:
|
[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. Zbl 1190.15024, MR 2551085 |
| Reference:
|
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| Reference:
|
[4] Hong, Y. P., Pan, C.-T.: A lower bound for the smallest singular value.Linear Algebra Appl. 172 27-32 (1992). Zbl 0768.15012, MR 1168494 |
| Reference:
|
[5] Landau, E.: Handbuch der Lehre von der Verteilung der Primzahlen. Erster Band.Leipzig u. Berlin: B. G. Teubner. X (1909). |
| Reference:
|
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| Reference:
|
[7] Tenenbaum, G.: Introduction à la Théorie Analytique et Probabiliste des Nombres.Cours Spécialisés 1 Paris: Société Mathématique de France (1995). Zbl 0880.11001, MR 1366197 |
| . |
