Geodesic
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
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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).
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).
DOI : 10.1007/s10587-016-0307-5
Classification : 11A05, 15A42, 15B36
Keywords: 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

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