Counting tuples restricted by pairwise coprimality conditions

Given a subset A of the set {1, . . . , v}2 we say that (a1, . . . , av) exhibits pairwise coprimality over A if gcd(ai, aj ) = 1 for all (i, j) ∈ A. For a given positive x and a given set A we give an asymptotic formula for the number of (a1, . . . , av) with 1 ≤ a1, . . . , av ≤ x that exhibit pairwise coprimality over A. This problem has been studied before by Hu.Ministerio de Economía y Competivida

The "visible" Wigner matrix

We consider the ``visible'' Wigner matrix, a Wigner matrix whose $(i, j)$-th entry is coerced to zero if $i, j$ are co-prime. Using a recent result from elementary number theory on co-primality patterns in integers, we show that the limiting spectral distribution of this matrix exists, and give explicit descriptions of its moments in terms of infinite products over primes $p$ of certain polynomials evaluated at $1/p$. We also consider the complementary ``invisible'' Wigner matrix.Comment: 12 pages, 5 figures, 1 tabl