报告题目：On the divisibility of matrices associated with multiplicative functions

报告人：洪绍方 教授（四川大学）

负责人：吴强

报告时间：2025年05月30日（星期五)上午11:00-12：00

报告地点：数学大楼914报告厅

参加人员：教师、研究生、本科生

摘要：Suppose that n and k are positive integers. Let S={x_1,...,x_n} be a sequence of n distinct positive integers, and let f be an integer-valued multiplicative function. The sequence S is called a divisor chain if x_{sigma(1)}|...|x_{sigma(n)} for some permutation sigma on {1,...,n}. We say that the sequence S consists of k coprime divisor chains if S can be partitioned as the union of k divisor chains S_1,...,S_k such that each element of S_i is coprime to each element of S_j for all integers i and j with 1 ≤ i ≠ j ≤ k . In this talk, we show that for any divisor chain S, the matrix (f(S)) with entries f(gcd(x_i,x_j)) divides the matrix (f[S]) with entries f(lcm(x_i, x_j)) in the ring M_n(Z) of n×n matrices over the integers if and only if f(min(S))|f(x_i) for all integers 1 ≤ i ≤ n . This strengthens a result of Hong obtained in 2003. For any positive integer a and any sequence S consisting of two coprime divisor chains with 1 Ï S, we show that the matrix (f(S^a)) divides the matrix (f[S^a]) in M_n(Z), where S^a:={x^a_1,...,x^a_n}. This confirms a conjecture of Chen and Hong. We show also that such factorization is no longer true in general if S consists of at least three

coprime divisor chains with 1 Ï S. We conjecture that if k ≥ 3, then the GCD matrix (S) does not divide the LCM matrix [S] in the ring M_k(Z) if S consists of the first k odd prime numbers.

报告人简介：洪绍方，四川大学数学学院教授、博士生导师，教育部新世纪优秀人才，四川省学术与技术带头人。1998年6月在四川大学获得理学博士学位，1998年7月至2000年6月，在中国科技大学数学系作博士后研究工作。2002年7月晋升教授。多次访问美国，法国，日本，以色列，韩国，以及香港和台湾等地区著名高校和研究所。于2013年参加在台湾大学举行的世界华人数学家大会，并作45分钟邀请报告。已经在国内外数学期刊发表学术论文百余篇，培养毕业硕士60多名，毕业博士20 多名，其中多人已晋升正高职称。