Lower bounds for the largest eigenvalue of the gcd matrix on
Czechoslovak Mathematical Journal (2016)
- Volume: 66, Issue: 3, page 1027-1038
- ISSN: 0011-4642
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topMerikoski, Jorma K.. "Lower bounds for the largest eigenvalue of the gcd matrix on $\lbrace 1,2,\dots ,n\rbrace $." Czechoslovak Mathematical Journal 66.3 (2016): 1027-1038. <http://eudml.org/doc/286799>.
@article{Merikoski2016,
abstract = {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).},
author = {Merikoski, Jorma K.},
journal = {Czechoslovak Mathematical Journal},
keywords = {eigenvalue bounds; greatest common divisor matrix},
language = {eng},
number = {3},
pages = {1027-1038},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Lower bounds for the largest eigenvalue of the gcd matrix on $\lbrace 1,2,\dots ,n\rbrace $},
url = {http://eudml.org/doc/286799},
volume = {66},
year = {2016},
}
TY - JOUR
AU - Merikoski, Jorma K.
TI - Lower bounds for the largest eigenvalue of the gcd matrix on $\lbrace 1,2,\dots ,n\rbrace $
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 3
SP - 1027
EP - 1038
AB - 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).
LA - eng
KW - eigenvalue bounds; greatest common divisor matrix
UR - http://eudml.org/doc/286799
ER -
References
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