Lower bounds for determinants of matrices associated with classes of arithmetical Functions

^∗supported by National Natural Science Foundation of china.

Abstract

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,… x_n } is a set of distinct positive integers and g(d) = h(d)ε R\{0} and f(d) > 0 whenever d|x for any x ε S, then

Furthermore, if whenever d|d ₁ for any d₁ d ₂ ε 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].

Keywords:

• The Gram - Schmidt orthogonalization process
• ged-closed
• the arithmetical function
• the quadratic function
• AMS Subject Classification (1991): 11A25, 11C20, 15A15

^∗supported by National Natural Science Foundation of china.

^∗supported by National Natural Science Foundation of china.

Notes

^∗supported by National Natural Science Foundation of china.