An upper bound for the norm of a GCD related matrix

(1)

A GCD-RELATED MATRIX

PENTTI HAUKKANEN

Received 10 November 2004; Revised 12 January 2005; Accepted 9 February 2005

We find an upper bound for the ᐉp norm of the n×n matrix whose i j entry is (i,j)s_/_[_i_,_j_]r_{, where (}_i_,_j_{) and [}_i_,_j_{] are the greatest common divisor and the least} com-mon multiple of iand j and where rand sare real numbers. In fact, we show that if

r >1/ pands < r−1/ p, then((i,j)s_/_[_i_,_j_]r₎

n×np< ζ(r p)2/ pζ(r p−sp)1/ p/ζ(2r p)1/ p for all positive integersn, whereζis the Riemann zeta function.

Copyright © 2006 Pentti Haukkanen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetS= {x1,x2,. . .,xn}be a set of distinct positive integers, and let f be an arithmetical function. Let (S)f denote then×nmatrix having f evaluated at the greatest common divisor (xi,xj) ofxiandxjas itsi jentry, that is, (S)f =(f((xi,xj))). Analogously, let [S]f denote then×nmatrix having f evaluated at the least common multiple [xi,xj] ofxi andxjas itsi jentry, that is, [S]f =(f([xi,xj])). The matrices (S)f and [S]f are referred to as the GCD and LCM matrix onSassociated withf respectively. Smith [12] calculated det(S)f whenSis a factor-closed set and det[S]f in a more special case. Since Smith a large number of results on GCD and LCM matrices have been presented in the literature. For general accounts see, for example, [3,5–8].

Norms of GCD matrices have not been studied much in the literature. Some results are obtained in [1,4], see also the references of [4] and [10, Chapter 3].

Letp∈Z+. Thepnorm of ann×nmatrixMis defined as

Mp=

_n

i=1 n

j=1 _m_{i j}p

1/ p

. (1.1)

(2)

Letr,s∈R. It is known [1, Theorem 3] that ifr >1/ p, then

lim n→∞

1 [i,j]r

n×n

p

= ζ(pr)3/ p

ζ(2pr)1/ p. (1.2)

We here generalize this result by showing that ifr >1/ pands < r−1/ p, then

lim n→∞

(i,j)s [i,j]r

n×n

p

=ζ(pr)2/ pζ(pr−ps)1/ p

ζ(2pr)1/ p , (1.3)

seeTheorem 3.1. This result also sharpens the rough estimation

(i,j)s [i,j]r

n×n

p

=O(1) (1.4)

given in [4, Theorem 3.1(3)].

2. Preliminaries

In this section we review the basic results on arithmetical functions needed in this paper. For more comprehensive treatments on arithmetical functions we refer to [2,9–11].

The Dirichlet convolutionf ∗gof two arithmetical functions f andgis defined as

(f ∗g)(n)= d|n

f(d)g(n/d). (2.1)

LetNu_,_u_∈_R_{, denote the arithmetical function defined as}_Nu₍_n₎₌_nu_{for all}_n_∈_Z+_, and letEdenote the arithmetical function defined asE(n)=1 for alln∈Z+_{. The Jordan} totient functionJk(n),k∈Z+, is defined as the number ofk-tuplesa1,a2,. . .,ak (modn) such that the greatest common divisor ofa1,a2,. . .,akandnis 1. By convention,Jk(1)=1. The M¨obius function μis the inverse ofE under the Dirichlet convolution. It is well known thatJk=Nk∗μ. This suggests we define

Ju(n)=Nu∗μ(n)=

d|n

duμ(n/d) (2.2)

for allu∈R. Sinceμis the inverse ofEunder the Dirichlet convolution, we have

nu=

d|n

Ju(d). (2.3)

An arithmetical function f is said to be multiplicative if f(1)=1 and

f(mn)=f(m)f(n) (2.4)

(3)

multiplicative. Each completely multiplicative function f distributes over the Dirichlet convolution, that is,

f(g∗h)=(f g)∗(f h) (2.5)

for all arithmetical functionsg and h. The inverse f−1 _{of a completely multiplicative} function f under the Dirichlet convolution is given as

f−1₌_{μ f .} _(2.6)

The Dirichlet series of an arithmetical function f is defined as

∞

n=1

f(n)

ns , (2.7)

where we assume (for brevity) thats is a real number. The Riemann zeta function is defined as

ζ(s)=

∞

n=1 1

ns, (2.8)

wheres >1. If the series ∞_n₌1f(n)/nsand ∞n=1g(n)/ns converge absolutely fors > s0, then

∞

n=1

f(n)

ns ∞

n=1

g(n)

ns = ∞

n=1

(f ∗g)(n)

ns (2.9)

and this last series converges absolutely fors > s0. Further, if the inverse f−1_of _f _under the Dirichlet convolution exists, then

∞

n=1

f−1₍_n₎

ns =

_∞

n=1

f(n)

ns

−1

(2.10)

and this series also converges absolutely fors > s0.

3. Results

Theorem3.1. Letr >1/ pands < r−1/ p. Then

lim n→∞

(i,j)s [i,j]r

n×n

p

=ζ(r p)2/ pζ(r p−sp)1/ p

ζ(2r p)1/ p . (3.1)

Proof. Denote

sn= n

i=1 n

j=1 (i,j)sp

(4)

Since (i,j)[i,j]=i j, we have for allp,r,s

sn= n

i=1 n

j=1

(i,j)(r+s)p

ir p_jr p . (3.3)

It is clear that

sn< ∞

i=1

∞

j=1

(i,j)(r+s)p

ir p_jr p . (3.4)

Making the change of variablesλ=(i,j),i=uλandj=vλ, we see that

sn< ∞

u=1

∞

v=1

∞

λ=1

λ(s−r)p ur p_vr p

= _∞

u=1 1

ur p

_∞

v=1 1

vr p

_∞

λ=1 1

λ(r−s)p

=ζ(r p)2ζ(r p−sp).

(3.5)

Note that all these series have only positive terms andr p,r p−sp >1. Thus,{sn}is in-creasing and bounded above, and so limn→∞sn=S exists. We deduce that the double series ((i,j)sp_/_[_i_,_j_]r p_{) converges absolutely, with sum}_S_.

We calculate the numberSas follows. We have

S=

∞

i=1

∞

j=1 (i,j)sp [i,j]r p =

∞

i=1

∞

j=1

(i,j)(r+s)p

ir p_jr p . (3.6)

From (2.3) we obtain

S=∞

i=1 1

ir p ∞

j=1 1

jr p

d|(i,j)

J(r+s)p(d)

=

∞

i=1 1

ir p

d|i

J(r+s)p(d)

1≤j<∞

j≡0 (modd)

1

jr p.

(3.7)

Sincer p >1, we can write

S=ζ(r p)

∞

i=1 1

ir p

d|i

J(r+s)p(d)

dr p . (3.8)

Since the function 1/dr p_{(i.e., the function}_N−r p_{) is completely multiplicative in}_d_{, on the} basis of (2.2) and (2.5) we have

S=ζ(r p)

∞

i=1 1

ir p

(5)

Since the function 1/ir p_{(i.e., the function}_N−r p_{again) is completely multiplicative in}_i_, on the basis of (2.5) again we have

S=ζ(r p)

∞

i=1

N−r p_∗_N−(r p−sp)_∗_μN−2r p₍_i₎_. _(3.10)

Sincer p,r p−sp >1, we can apply (2.6)–(2.10) to obtain

S=ζ(r p)ζ(r p)ζ(r p−sp)/ζ(2r p). (3.11)

This completes the proof ofTheorem 3.1.

Corollary3.2. Letr >1/ pands < r−1/ p. Then, for alln∈Z+_,

(i,j)s [i,j]r

n×n

p

<ζ(r p)

2/ p_ζ₍_{r p}₋_sp₎1/ p

ζ(2r p)1/ p . (3.12)

The spectral norm of ann×nmatrixMis defined as

MS=max

λ:λis an eigenvalue ofM∗M. (3.13)

Corollary3.3. Letr >1/2ands < r−1/2. Then, for alln∈Z+_,

(i,j)s [i,j]r

n×n

S

<ζ(2r)ζ

2(r−s)1/2

ζ(4r)1/2 . (3.14)

Proof. It is known thatMS≤ M2. ThusCorollary 3.3 follows from Corollary 3.2.

Acknowledgment

The author wishes to thank the anonymous referees for valuable comments, which led on to an improvement ofTheorem 3.1.

References

[1] E. Altinisik, N. Tuglu, and P. Haukkanen,A note on bounds for norms of the reciprocal LCM matrix, Mathematical Inequalities & Applications7(2004), no. 4, 491–496.

[2] T. M. Apostol,Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer, New York, 1976.

[3] K. Bourque and S. Ligh, Matrices associated with classes of arithmetical functions, Journal of Number Theory45(1993), no. 3, 367–376.

[4] P. Haukkanen,On thepnorm of GCD and related matrices, JIPAM. Journal of Inequalities in

Pure and Applied Mathematics5(2004), no. 3, Article 61, 7 pp.

[5] P. Haukkanen and J. Sillanp¨a¨a,Some analogues of Smith’s determinant, Linear Multilinear Algebra

41(1996), no. 3, 233–244.

(6)

[7] S. Hong,GCD-closed sets and determinants of matrices associated with arithmetical functions, Acta Arithmetica101(2002), no. 4, 321–332.

[8] I. Korkee and P. Haukkanen,On meet and join matrices associated with incidence functions, Linear Algebra and its Applications372(2003), 127–153.

[9] P. J. McCarthy,Introduction to Arithmetical Functions, Universitext, Springer, New York, 1986. [10] J. S´andor and B. Crstici,Handbook of Number Theory, II, Springer, New York, 2004.

[11] R. Sivaramakrishnan,Classical Theory of Arithmetic Functions, Monographs and Textbooks in Pure and Applied Mathematics, vol. 126, Marcel Dekker, New York, 1989.

[12] H. J. S. Smith,On the value of a certain arithmetical determinant, Proceedings of the London Mathematical Society7(1875/1876), 208–212.

Pentti Haukkanen: Department of Mathematics, Statistics and Philosophy, University of Tampere, 33014 Tampere, Finland