Determinants of Generalized GCD Matrices Associated with Arithmetic Functions

ZHU Yuqing, LIAN Dongyan, DIAO Tianbo, HU Shuangnian,2

( 1. College of Mathematics and Statistics, Nanyang Institute of Technology, Nanyang 473004, Henan;2. College of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, Henan)

1 Introduction and statements of main results

Throughout this section, we letfbe an arithmetic function andS={x1,x2,…,xn} be a set ofndistinct positive integers. We can now give the first two main results of this paper, which extend Bege’s results[20].

Then each of the following is true:

and then×nmatrixD=(dij) is defined by

In what follows, we always let then×nmatricesCandDbe defined as in Theorem 1.1.From Theorem 1.1, one can deduce the following result of Bege[20].

Then each of the following is true:

FromTheorem1.2,onecandeducethefollowingresultofBege[20].

From Theorems 1.1 and 1.2, we can easily get the following result.

We organize this paper as follows. In Section 2, we prove Theorems 1.1 and 1.2. In Section 3, some examples are given to illustrate our main results.

2 Proof of Theorems 1.1 and 1.2

In this section, we prove Theorems 1.1-1.2. We begin with the proof of Theorem 1.1.

ProofofTheorem1.1(i) Write

A=Cdiag(f(x1),f(x2),…,f(xn))DT.

Then for any integersiandj(1≤i,j≤n), we have

Thus,

So the desired result follows immediately. This completes the proof of part (i).

det(C)det(diag(f(x1),f(x2),…,f(xn)))×

(iii) As the argument given in part (ii), we let 1≤x1

This ends the proof of Theorem 1.1.

ProofofTheorem1.2(i) For any integersiandjwith 1≤i,j≤n, we have

So the desired result follows immediately. This completes the proof of part (i).

(ii) Using part (i), one infers that

det(D)det(diag(f(x1),f(x2),…,f(xn)))×

Since Corollaries 1.1～1.3 are very easy to get, we omit their proofs here.

3 Examples

In this section, we give some examples to demonstrate our main results.

Example3.1LetS={2,4,8,12,16} andλbe the Liouville function which is defined by

Then one has

By Theorems 1.1 and 1.2, we have

and

Furthermore, we have

Example3.2LetS={2,4,5,8}. For any positive integern, we letf(n)=n. Then we obtain

From Theorems 1.1 and 1.2, we have

and

Moreover, we have

and

AcknowledgementsThe authors would like to thank the anonymous referee for careful reading of the manuscript and helpful comments that improve the presentation of this paper.

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