Lower bounds for determinants of matrices associated with classes of arithmetical Functions
Lower bounds for determinants of matrices associated with classes of arithmetical Functions
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DOI：
10.1080/03081089908818599
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发表时间：
1999-04
影响因子：
1.1
通讯作者：
Shaofang Hong
中科院分区：
文献类型：
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作者：
Shaofang Hong
设f、g和h是数论函数。定义两个变量t和r的算术函数f（t，r）如下：本文证明了：如果S = {x1，x2，. xn }是一组不同的正整数，g（d）= h（d）eR 0}，且f（d）> 0，则当d| x对于任意x e S，则此外，如果每当d|对于任意d1 d2 e S满足d1| d2 }，则等式成立当且仅当S是gcd闭的。这个结果改进了Bourque和Ligh的一个定理[J. NumerTheory 45（1993），367-376]。
Let f and g and h be arithmetical functions. Define an arithmetical function ψ(t,r) of two variables t and r as follows: . In this paper we show that if S = {x1,x2,… xn } is a set of distinct positive integers and g(d) = h(d)e R\{0} and f(d) > 0 whenever d|x for any x e S, then Furthermore, if whenever d|d 1 for any d1 d 2 e S satisfying d1|d2 },then the equality holds if and only if S is gcd-closed. This result improves a theorem of Bourque and Ligh [J. Number Theory 45 (1993), 367–376].