Lower bounds for the largest eigenvalue of the gcd matrix on $\{1,2,\dots ,n\}$

Merikoski, Jorma K.

doi:10.1007/s10587-016-0307-5

Geodesic

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Lower bounds for the largest eigenvalue of the gcd matrix on

{1, 2, \dots, n}

Merikoski, Jorma K.

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 1027-1038.

Voir la notice de l'article dans Czech Digital Mathematics Library

Résumé

Consider the

n \times n

matrix with

(i, j)

'th entry

gcd (i, j)

. Its largest eigenvalue

λ_{n}

and sum of entries

s_{n}

satisfy

λ_{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

λ_{n} > 6 π^{- 2} n \log n

for all

n

. If

n

is large enough, this follows from F. Balatoni (1969).

MR Zbl

DOI : 10.1007/s10587-016-0307-5

Classification : 11A05, 15A42, 15B36
Mots-clés : eigenvalue bounds; greatest common divisor matrix

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Merikoski, Jorma K. Lower bounds for the largest eigenvalue of the gcd matrix on $\{1,2,\dots ,n\}$. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 1027-1038. doi : 10.1007/s10587-016-0307-5. https://geodesic-test.mathdoc.fr/articles/10.1007/s10587-016-0307-5/

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