| Title:
|
Lower bounds for the largest eigenvalue of the gcd matrix on $\{1,2,\dots ,n\}$ (English) |
| Author:
|
Merikoski, Jorma K. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
3 |
| Year:
|
2016 |
| Pages:
|
1027-1038 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Consider the $n\times n$ matrix with $(i,j)$'th entry $\gcd {(i,j)}$. Its largest eigenvalue $\lambda _n$ and sum of entries $s_n$ satisfy $\lambda _n>s_n/n$. Because $s_n$ cannot be expressed algebraically as a function of $n$, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S. Hong, R. Loewy (2004). We also conjecture that $\lambda _n>6\pi ^{-2}n\log {n}$ for all $n$. If $n$ is large enough, this follows from F. Balatoni (1969). (English) |
| Keyword:
|
eigenvalue bounds |
| Keyword:
|
greatest common divisor matrix |
| MSC:
|
11A05 |
| MSC:
|
15A42 |
| MSC:
|
15B36 |
| idZBL:
|
Zbl 06644048 |
| idMR:
|
MR3556882 |
| DOI:
|
10.1007/s10587-016-0307-5 |
| . |
| Date available:
|
2016-10-01T15:45:42Z |
| Last updated:
|
2023-10-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145886 |
| . |
| Reference:
|
[1] Altınışık, E., Keskin, A., Yıldız, M., Demirbüken, M.: On a conjecture of Ilmonen, Haukkanen and Merikoski concerning the smallest eigenvalues of certain GCD related matrices.Linear Algebra Appl. 493 (2016), 1-13. Zbl 1334.15079, MR 3452722 |
| Reference:
|
[2] Balatoni, F.: On the eigenvalues of the matrix of the Smith determinant.Mat. Lapok 20 (1969), 397-403 Hungarian. Zbl 0213.32303, MR 0291186 |
| Reference:
|
[3] Beslin, S., Ligh, S.: Greatest common divisor matrices.Linear Algebra Appl. 118 (1989), 69-76. Zbl 0672.15005, MR 0995366 |
| Reference:
|
[4] Hong, S., Loewy, R.: Asymptotic behavior of eigenvalues of greatest common divisor matrices.Glasg. Math. J. 46 (2004), 551-569. Zbl 1083.11021, MR 2094810, 10.1017/S0017089504001995 |
| Reference:
|
[5] Horn, R. A., Johnson, C. R.: Matrix Analysis.Cambridge University Press, Cambridge (2013). Zbl 1267.15001, MR 2978290 |
| Reference:
|
[6] Mitrinović, D. S., Sándor, J., Crstici, B.: Handbook of Number Theory.Mathematics and Its Applications 351 Kluwer Academic Publishers, Dordrecht (1995). Zbl 0862.11001, MR 1374329 |
| Reference:
|
[7] Smith, H. J. S.: On the value of a certain arithmetical determinant.Proc. L. M. S. 7 208-213 (1875). MR 1575630 |
| Reference:
|
[8] Tóth, L.: A survey of gcd-sum functions.J. Integer Seq. (electronic only) 13 (2010), Article ID 10.8.1, 23 pages. Zbl 1206.11118, MR 2718232 |
| Reference:
|
[9] Yaglom, A. M., Yaglom, I. M.: Non-elementary Problems in an Elementary Exposition.Gosudarstv. Izdat. Tehn.-Teor. Lit., Moskva (1954), Russian. MR 0070671 |
| Reference:
|
[10] Weisstein, E. W.: Faulhaber's Formula.From Mathworld---A Wolfram Web Resource, http://mathworld.wolfram.com/FaulhabersFormula.html. |
| . |
