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摘要
Let S = {x(1), ... , x(n)} be a set of n distinct positive integers. For x, y is an element of S and y < x, we say the y is a greatest-type divisor of x in S if y vertical bar x and it can be deduced that z = y from y vertical bar z, z vertical bar x, z < x and z is an element of S. For x is an element of S, let G(S) (x) denote the set of all greatest-type divisors of x in S. For any arithmetic function f, let (f (x(i), x(j))) denote the n x n matrix having f evaluated at the greatest common divisor (x(i), x(j)) of x(i), and x(j) as its i,j-entry and let (f[x(i), x(j)]) denote the n x n matrix having f evaluated at the least common multiple [x(i), x(j)] of x(i) and x(j) as its i,j-entry. In this paper, we assume that S is a gcd-closed set and max(x is an element of S) {vertical bar G(S) (x)vertical bar} = 1. We show that if f is a multiplicative function such that (f * mu)(d) is an element of Z whenever d vertical bar 1cm(S) and f (a) vertical bar f (b) whenever a vertical bar b and a, b is an element of S and (f[x(i), x(j))) is nonsingular, then the matrix (f (x(i), x(j))) divides the matrix (f (x(i), x(j))) in the ring M(n)(Z) of n x n matrices over the integers. As a consequence, we show that (f (x(i), x(j))) divides (f[x(i), x(j))) in the ring M(n) (Z) if (f * mu)(d) is an element of Z whenever d vertical bar 1cm(S) and! is a completely multiplicative function such that (f (x(i), x(j))) is nonsingular. This confirms a conjecture of Hong raised in 2004.
- 出版日期2011-10-28
- 单位四川大学; 攀枝花学院
