ASYMPTOTIC BEHAVIOR OF LARGEST EIGENVALUE OF MATRICES ASSOCIATED WITH COMPLETELY EVEN FUNCTIONS (MOD r)

Abstract

Given an arbitrary strictly increasing infinite sequence [Formula: see text] of positive integers, let Sn = {x1,…, xn} for any integer n ≥ 1. Let q ≥ 1 be a given integer and f an arithmetical function. Let [Formula: see text] be the eigenvalues of the matrix (f(xi, xj)) having f evaluated at the greatest common divisor (xi, xj) of xi and xj as its i, j-entry. We obtain a lower bound depending only on x1 and n for [Formula: see text] if (f * μ)(d) < 0 whenever d|x for any x ∈ Sn, where g * μ is the Dirichlet convolution of f and μ. Consequently we show that for any sequence [Formula: see text], if f is a multiplicative function satisfying that f(pk) → ∞ as pk → ∞, f(2) > 1 and f(pm) ≥ f(2)f(pm−1) for any prime p and any integer m ≥ 1 and (f *μ)(d) > 0 whenever d|x for any [Formula: see text], then [Formula: see text] approaches infinity when n goes to infinity.