ON THE MAXIMUM ROW AND COLUMN SUM NORM OF GCD AND RELATED MATRICES

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Norm of GCD and Related Matrices Pentti Haukkanen vol. 8, iss. 4, art. 97, 2007

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ON THE MAXIMUM ROW AND COLUMN SUM NORM OF GCD AND RELATED MATRICES

PENTTI HAUKKANEN

Department of Mathematics, Statistics and Philosophy, FIN-33014 University of Tampere,

Finland

EMail:mapehau@uta.fi

Received: 11 July, 2007

Accepted: 27 October, 2007 Communicated by: L. Tóth

2000 AMS Sub. Class.: 11C20; 15A36; 11A25.

Key words: GCD matrix, LCM matrix, Smith’s determinant, Maximum row sum norm, Max- imum column sum norm,O-estimate.

Abstract: We estimate the maximum row and column sum norm of then×nmatrix, whose ijentry is(i, j)^s/[i, j]^r, wherer, s∈R,(i, j)is the greatest common divisor of iandjand[i, j]is the least common multiple ofiandj.

Acknowledgements: The author wishes to thank Pauliina Ilmonen for calculations which led to Re- marks1–3.

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1. Introduction

LetS ={x₁, x₂, . . . , x_n}be a set of distinct positive integers, and letf be an arithmetical function. Let(S)_f denote then×nmatrix havingfevaluated at the greatest common divisor (x_i, x_j)of x_i and x_j as itsij entry, that is, (S)_f = (f((x_i, x_j))).

Analogously, let[S]f denote then×n matrix havingf evaluated at the least common multiple [xi, xj]of xi and xj as its ij entry, that is,[S]f = (f([xi, xj])). The matrices(S)_f and[S]_f are referred to as the GCD and LCM matrices onS associated withf. H. J. S. Smith [15] calculateddet(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 e.g.

[8,9,12,14].

Norms of GCD matrices have not been discussed much in the literature. Some results for the`_pnorm are reported in [1,6,7], see also the references in [6]. In this paper we consider the maximum row sum norm in a similar way as we considered the`_pnorm in [6]. Since the matrices in this paper are symmetric, all the results also hold for the maximum column sum norm.

The maximum row sum norm of ann×nmatrixM is defined as

|||M|||_∞= max

1≤i≤n n

X

j=1

|m_ij|.

Letr, s∈R. LetAdenote then×nmatrix, whosei, jentry is given as

(1.1) aij = (i, j)^s

[i, j]^r,

where(i, j)is the greatest common divisor ofiandj and[i, j]is the least common multiple ofi andj. For s = 1, r = 0and s = 0, r = −1, respectively, the matrix

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A is the GCD and the LCM matrix on{1,2, . . . , n}. Fors = 1, r = 1 the matrix A is the Hadamard product of the GCD matrix and the reciprocal LCM matrix on {1,2, . . . , n}. In this paper we estimate the maximum row sum norm of the matrix Agiven in(1.1)for allr, s∈R.

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2. Preliminaries

In this section we review the basic results on arithmetical functions needed in this paper. For more comprehensive treatments of arithmetical functions we refer to [2,13,14].

The Dirichlet convolution f ∗g of two arithmetical functionsf andg is defined as

(f ∗g)(n) =X

d|n

f(d)g(n/d).

Let Nû, u ∈ R, denote the arithmetical function defined as Nû(n) = nû for all n ∈ Z⁺, and let E denote the arithmetical function defined as E(n) = 1 for all n∈Z⁺. The divisor functionσu,u∈R, is defined as

(2.1) σ_u(n) =X

d|n

d^u = (N^u∗E)(n).

It is known that if0≤u <1, then

(2.2) σ_u(n) = O(n^u+)

for all >0(see [5]),

(2.3) σ1(n) =O(nlog logn)

(see [4,11,13]), and ifu >1, then

(2.4) σ_u(n) =O(n^u)

(see [3,4,13]).

The Jordan totient function J_k(n), k ∈ Z⁺, is defined as the number ofk-tuples a1, a2, . . . , ak (mod n)such that the greatest common divisor ofa1, a2, . . . , akand

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nis1. By convention,J_k(1) = 1. The Möbius functionµis the inverse ofE under the Dirichlet convolution. It is well known that J_k = N^k ∗µ. This suggests we defineJ_u = N^u∗µfor allu ∈ R. Since µis the inverse ofE under the Dirichlet convolution, we have

(2.5) n^u =X

d|n

J_u(d).

It is easy to see that

J_u(n) = n^uY

p|n

(1−p^−u).

We thus have

(2.6) 0≤Ju(n)≤n^u foru≥0.

The following estimates for the summatory function ofN^u are well known (see [2]):

X

k≤n

k^−u =O(1) ifu >1, (2.7)

X

k≤n

k⁻¹ =O(logn), (2.8)

X

k≤n

k^−u =O(n^1−u) ifu <1.

(2.9)

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3. Results

In Theorems 3.1 – 3.5 we estimate the maximum row sum norm of the matrix A given in(1.1). Their proofs are based on the formulas in Section2and the following observations.

Since(i, j)[i, j] =ij, we have for allr, s (3.1) |||A|||_∞= max

1≤i≤n n

X

j=1

(i, j)^s

[i, j]^r = max

1≤i≤n n

X

j=1

(i, j)^r+s i^rj^r .

From(2.5)we obtain

|||A|||_∞= max

1≤i≤n

1 i^r

n

X

j=1

1 j^r

X

d|(i,j)

Jr+s(d)

= max

1≤i≤n

1 i^r

X

d|i

J_r+s(d)

n

X

j=1 d|j

1 j^r

= max

1≤i≤n

1 i^r

X

d|i

J_r+s(d) d^r

[n/d]

X

j=1

1 j^r. (3.2)

Theorem 3.1. Suppose thatr >1.

1. Ifs≥r, then|||A|||_∞=O(n^s−r).

2. Ifs < r, then|||A|||_∞ =O(1).

Proof. Letr >1ands≥0. Then, by (3.2) and(2.7),

|||A|||_∞ =O(1) max

1≤i≤n

1 i^r

X

d|i

J_r+s(d) d^r .

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Sincer+s≥0, according to(2.6)and(2.1),

|||A|||_∞=O(1) max

1≤i≤n

σ_s(i) i^r . Now, ifs ≥r >1, then on the basis of(2.4),

|||A|||_∞ =O(1) max

1≤i≤ni^s−r =O(n^s−r).

If0≤s < r, then

|||A|||_∞=O(1) max

1≤i≤ni^s−r+ =O(1).

Letr >1ands <0. Then

|||A|||_∞ ≤ max

1≤i≤n n

X

j=1

1

j^r =O(1).

Theorem 3.2. Suppose thatr= 1.

1. Ifs >1, then|||A|||_∞=O(n^s−1logn).

2. Ifs= 1, then|||A|||_∞=O(logn log logn).

3. Ifs <1, then|||A|||_∞=O(logn).

Proof. From (3.2) withr= 1we obtain

|||A|||_∞= max

1≤i≤n

1 i

X

d|i

J_s+1(d) d

[n/d]

X

j=1

1 j.

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By(2.8),

|||A|||_∞ =O(logn) max

1≤i≤n

1 i

X

d|i

J_s+1(d)

d .

Sinces≥0, on the basis of(2.6)and(2.1),

|||A|||_∞=O(logn) max

1≤i≤n

σ_s(i) i . Ifs >1, then according to(2.4),

|||A|||_∞=O(logn) max

1≤i≤ni^s−1 =O(n^s−1logn).

Ifs= 1, then according to(2.3),

|||A|||_∞=O(logn)O(log logn) = O(logn log logn).

If0≤s <1, then according to(2.2),

|||A|||_∞ =O(logn) max

1≤i≤ni^s−1+ =O(logn).

Ifs <0, then according to(3.1),

|||A|||_∞≤ max

1≤i≤n n

X

j=1

1

j =O(logn).

Remark 1. LetkMk₁denote the sum norm (or`₁ norm) of ann×nmatrixM, that is

kMk₁ =

n

X

i=1 n

X

j=1

|m_ij|.

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It is known [6, Theorem 3.2(1)] that (3.3)

(i, j)^s/[i, j]

1 =O(n^slog²n), s≥1.

SincekMk₁ ≤n|||M|||_∞for alln×nmatricesM (see [10]), we obtain from Theo- rem3.2(1,2) an improvement on(3.3)as

(i, j)^s/[i, j]

1

=O(n^slogn), s >1, (3.4)

(i, j)/[i, j]

1

=O(nlogn log logn).

(3.5)

Theorem 3.3. Suppose thatr <1.

1. Ifs >2−r, then|||A|||_∞=O(n^s−r).

2. Ifs= 2−r, then|||A|||_∞=O(n^2−2rlog logn).

3. Ifmax{1−r,1} ≤s <2−r, then|||A|||_∞ =O(n^s−r+)for all >0.

4. If1−r ≤s <1, then|||A|||_∞=O(n^1−r).

Proof. Letr <1. By (3.2) and(2.9),

|||A|||_∞=O(n^1−r) max

1≤i≤n

1 i^r

X

d|i

J_r+s(d)

d .

Sincer+s≥0, by(2.6)and(2.1),

|||A|||_∞ =O(n^1−r) max

1≤i≤n

σ_r+s−1(i) i^r .

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Ifs >2−rorr+s−1>1, then according to(2.4),

|||A|||_∞=O(n^1−r) max

1≤i≤ni^s−1. Sinces−1≥0, we have

|||A|||_∞ =O(n^s−r).

Ifs= 2−rorr+s−1 = 1, then according to(2.3),

|||A|||_∞=O(n^1−r) max

1≤i≤ni^1−rlog logi.

Since1−r >0, we have

|||A|||_∞=O(n^2−2rlog logn).

If1−r≤s <2−ror0≤r+s−1<1, then according to(2.2), (3.6) |||A|||_∞ =O(n^1−r) max

1≤i≤ni^s−1+.

If s ≥ 1 in (3.6), we obtain |||A|||_∞ = O(n^s−r+). If s < 1 in (3.6), we obtain

|||A|||_∞=O(n^1−r).

Corollary 3.4. Suppose thatr= 0.

1. Ifs >2, then|||A|||_∞=O(n^s).

2. Ifs= 2, then|||A|||_∞=O(n²log logn).

3. If1≤s <2, then|||A|||_∞ =O(n^s+)for all >0. In particular, fors= 1, (3.7)

(i, j)

_∞=O(n¹⁺)f or all >0.

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Remark 2. LetkMk₂ denote the`₂ norm of ann×nmatrixM, that is kMk₂ =

n

X

i=1 n

X

j=1

m²_ij.

It is known [6, Theorem 3.2(1)] that (3.8)

(i, j)^3/2/[i, j]^1/2

2 =O(n^3/2logn).

Since kMk₂ ≤ √

n|||M|||_∞ for all n ×n matrices M (see [10]), we obtain from Theorem3.3(2) an improvement on(3.8)as

(3.9)

(i, j)^3/2/[i, j]^1/2

2 =O(n^3/2log logn).

In Theorem 3.5 we treat the remaining cases ofr and s in the most elementary way.

Theorem 3.5.

1. If0≤r <1ands≤0, then|||A|||_∞ =O(n^1−r).

2. Ifr <0ands ≤0, then|||A|||_∞=O(n^1−2r).

3. If0≤r <1,s >0andr+s <1, then|||A|||_∞=O(n^1+s−r).

4. Ifr <0,s >0andr+s <1, then|||A|||_∞=O(n^1+s−2r).

Proof. Let0≤r <1ands ≤0. Then, according to (3.1) and(2.9)

|||A|||_∞≤ max

1≤i≤n n

X

j=1

1

j^r =O(n^1−r).

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Letr <0ands≤0. Then, according to (3.1) and the inequality[i, j]< n²

|||A|||_∞≤ max

1≤i≤n n

X

j=1

[i, j]^−r < max

1≤i≤n n

X

j=1

n^−2r =O(n^1−2r).

Let0≤r <1,s >0andr+s <1. Then, according to (3.1) and(2.9)

|||A|||_∞≤n^s max

1≤i≤n n

X

j=1

1

j^r =O(n^1+s−r).

Let r < 0, s > 0 and r + s < 1. Then, according to (3.1) and the inequality [i, j]< n²

|||A|||_∞≤ max

1≤i≤n n

X

j=1

n^s

n^2r =O(n^1+s−2r).

Remark 3. Applying [6, Theorem 3.3] and the inequality|||M|||_∞ ≤√

nkMk₂for all n×nmatricesM (see [10]) a partial improvement on Theorem3.5(4) of the present paper as

(a) ifr <0,s >0and1/2< r+s <1, then|||A|||_∞=O(n^1+s−r), (b) ifr <0,s >0andr+s= 1/2, then|||A|||_∞=O(n^−2r+3/2log^1/2n), (c) ifr <0,s >1/2andr+s <1/2, then|||A|||_∞ =O(n^−2r+3/2).

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