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

Title:

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

Creator:

Merikoski, Jorma K

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:09b6f243-893a-42f1-9e36-61549cd0a302
uuid:09b6f243-893a-42f1-9e36-61549cd0a302
issn:0011-4642

Subject:

matematika, mathematics, eigenvalue bounds, greatest common divisor matrix, 13, and 51

Type:

model:article and TEXT

Format:

print, bez média, and svazek

Description:

Consider the n×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 sn 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)., Jorma K. Merikoski., and Obsahuje seznam literatury

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

1027-1038

Source:

Czechoslovak Mathematical Journal | 2016 Volume:66 | Number:3

Harvested from:

CDK

Metadata only:

false