Previous |  Up |  Next


eigenvalue bounds; greatest common divisor matrix
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).
[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. MR 3452722 | Zbl 1334.15079
[2] Balatoni, F.: On the eigenvalues of the matrix of the Smith determinant. Mat. Lapok 20 (1969), 397-403 Hungarian. MR 0291186 | Zbl 0213.32303
[3] Beslin, S., Ligh, S.: Greatest common divisor matrices. Linear Algebra Appl. 118 (1989), 69-76. MR 0995366 | Zbl 0672.15005
[4] Hong, S., Loewy, R.: Asymptotic behavior of eigenvalues of greatest common divisor matrices. Glasg. Math. J. 46 (2004), 551-569. DOI 10.1017/S0017089504001995 | MR 2094810 | Zbl 1083.11021
[5] Horn, R. A., Johnson, C. R.: Matrix Analysis. Cambridge University Press, Cambridge (2013). MR 2978290 | Zbl 1267.15001
[6] Mitrinović, D. S., Sándor, J., Crstici, B.: Handbook of Number Theory. Mathematics and Its Applications 351 Kluwer Academic Publishers, Dordrecht (1995). MR 1374329 | Zbl 0862.11001
[7] Smith, H. J. S.: On the value of a certain arithmetical determinant. Proc. L. M. S. 7 208-213 (1875). MR 1575630
[8] Tóth, L.: A survey of gcd-sum functions. J. Integer Seq. (electronic only) 13 (2010), Article ID 10.8.1, 23 pages. MR 2718232 | Zbl 1206.11118
[9] Yaglom, A. M., Yaglom, I. M.: Non-elementary Problems in an Elementary Exposition. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moskva (1954), Russian. MR 0070671
[10] Weisstein, E. W.: Faulhaber's Formula. From Mathworld---A Wolfram Web Resource,
Partner of
EuDML logo