Lower bounds for the largest eigenvalue of the gcd matrix on {1, 2,..., n}
Jorma Kaarlo Merikoski · Czechoslovak Mathematical Journal · 2016
Consider the n×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 > 6π−2 nlogn for all n. If n is large enough, this follows from F.Balatoni (1969).