Let S={x_1,x_2,...,x_n } be a set of n distinct positive integers and f be an arithmetic function.By（f[S]）（resp.（ f[S]））,we denote the n＊n matrix whose i,j entry is Σ[x_i,x_j]｜l l∈S f（l） （resp.Σx∈Sf（x...Let S={x_1,x_2,...,x_n } be a set of n distinct positive integers and f be an arithmetic function.By（f[S]）（resp.（ f[S]））,we denote the n＊n matrix whose i,j entry is Σ[x_i,x_j]｜l l∈S f（l） （resp.Σx∈Sf（x）-Σ x_i,｜l l∈S f（l）-Σ x_j,｜l l∈S f（l）＋Σ[x_i,x_j]｜l l∈S f（l））.In this paper,we first investigate the structures of the matrices （ f[S]） and（ f[S]）,then we give the formulae for the determinants of these matrices.These extend the results obtained by Bege in 2011.Finally,we give two examples to demonstrate the validity of our main results.展开更多
基金Supported partially by the National Natural Science Foundation of China(11501387)Key Program of Universities of Henan Province of China(17A110010)+1 种基金China Postdoctoral Science Foundation Funded Project(2016M602251)the Natural Science Foundation of Henan Province(162300410076)
文摘Let S={x_1,x_2,...,x_n } be a set of n distinct positive integers and f be an arithmetic function.By（f[S]）（resp.（ f[S]））,we denote the n＊n matrix whose i,j entry is Σ[x_i,x_j]｜l l∈S f（l） （resp.Σx∈Sf（x）-Σ x_i,｜l l∈S f（l）-Σ x_j,｜l l∈S f（l）＋Σ[x_i,x_j]｜l l∈S f（l））.In this paper,we first investigate the structures of the matrices （ f[S]） and（ f[S]）,then we give the formulae for the determinants of these matrices.These extend the results obtained by Bege in 2011.Finally,we give two examples to demonstrate the validity of our main results.
