与一类半乘法函数相关联矩阵的行列式的界

Bounds for determinants of Matrices Associated with Classes of Semi-multiplicative Functions

设f是算术函数,S={x1,x2,…,xn}是一个n元正整数集.(f[xi,xj])表示一个n阶方阵,它的i行j列处的元素为函数f在[xi,xj]处的取值,其中[xi,xj]为xi和xi的最小公倍数.作者证明了对于某个算术函数类,若f是一个半乘法函数且1f属于这个函数类,则矩阵(f[xi,xj])是半正定的,进而给出了其行列式的明确的下界和上界.若以f(c)表示函数f的c重狄利克雷乘积,则矩阵1f(c)[xi,xj]也有类似的结论.

Let f be an arithmetical function and S = {x_l,...,x_n} be a set of distinct positive integers. Let ((f)) denote the n×n matrix having f evaluated at the least common multiple of x_i and x_j as its i,j entry. The authors show that for a certain class of arithmetical functions, the matrix ((f)) is semi-positive definite. They also get sharp lower and upper bounds for det(f). Denot the c-th Dirichlet convolution of f by f^((c)), they show that the n×n matrix (1f^((c))) has similar results.