Subgroups of rank three groups

Now we consider the subgroups of the groupZ^mˆZⁿˆZ^r, which has rank three when pm, n, rq ą 1.

Theorem 2.8.5 (Hampejs and T´oth [38, Th. 2.1]). Let m, n, r P N. The subgroups of the group Γ“Z^mˆZⁿˆZ^r can be represented as follows.

(i) Choose a, b, cPN such that a|m, b|n, c|r.

(ii) Compute A:“gcdpa, n{bq, B :“gcdpb, r{cq, C :“gcdpa, r{cq.

(iii) Compute

X:“ ABC

gcdpapr{cq, ABCq. (iv) Let s :“at{A, where 0ďtďA´1.

(v) Let

v :“ bX

Bgcdpt, Xqw, where 0ďwďBgcdpt, Xq{X´1.

(vi) Find a solution u₀ of the linear congruence

pr{cqu”rvs{pbcq (mod a).

(vii) Let u:“u₀`az{C, where 0ďz ďC´1.

(viii) Consider

Ua,b,c,t,w,z :“ xpa,0,0q,ps, b,0q,pu, v, cqy

“ tpia`js`ku, jb`kv, kcq: 0ďiďn{a´1,0ďj ďn{b´1,0ďk ďn{c´1u.

Then Ua,b,c,t,w,z is a subgroup of order mnr{pabcq of Γ. Moreover, there is a bijection between the set of6-tuples pa, b, c, t, w, zq satisfying the conditions (i)-(viii) and the set of subgroups of Γ.

Next we give a formula for the number of subgroups of Γ.

Theorem 2.8.6 (Hampejs and T´oth [38, Th. 2.2]). For every m, n, r P N the total number of subgroups of the group ZmˆZnˆZr is given by

spm, n, rq “ ÿ

a|m, b|n, c|r

ABC

X² PpXq, (2.47)

with the notation of Theorem 2.8.5, where Ppnq is the gcd-sum function.

If one of m, n, r is 1, then formula (2.47) reduces to (2.44).

I remark that in the joint paper with T˘arn˘auceanu [93, Cor. 2.2] we proved, based on formula (2.38), that the total number of subgroups of the rank three p-group Zp^λ1 ˆ Zp^λ2 ˆZp^λ3, withλ₁ ěλ₂ ěλ₃ ě1, is given by the following polynomial:

Fppq pp²´1q²pp´1q,

where

Fppq “pλ₃`1qpλ<sub>1</sub>´λ<sub>2</sub>`1qp^λ2`λ3`5`2pλ<sub>3</sub>`1qp^λ2`λ3`4

´2pλ₃`1qpλ<sub>1</sub>´λ<sub>2</sub>qp<sup>λ2`λ3`3</sup>´2pλ<sub>3</sub>`1qp^λ2`λ3`2

` pλ<sub>3</sub>`1qpλ₁´λ₂´1qp^λ2`λ3`1´ pλ₁`λ<sub>2</sub>´λ<sub>3</sub>`3qp^2λ3`4

´2p^{2λ3`3</sup>` pλ1`λ2´λ3 ´1qp<sup>2λ3`2}

` pλ<sub>1</sub>`λ₂`λ<sub>3</sub>`5qp² `2p´ pλ<sub>1</sub> `λ₂ `λ<sub>3</sub>`1q.

This formula was also obtained by Oh [72, Cor. 2.2], using different arguments.

The final result of this chapter is an asymptotic formula for the number of subgroups spn, n, nq of Z³n. The values of this function for 1 ď n ď 50, obtained by Theorem 2.8.6 and using the software Mathematica, are given in Table 1.

Table 1. Values ofspnq:“spn, n, nq for 1ďn ď50

n spnq n spnq n spnq n spnq n spnq

1 1 11 268 21 3248 31 1988 41 3448

2 16 12 3612 22 4288 32 22308 42 51968

3 28 13 368 23 1108 33 7504 43 3788

4 129 14 1856 24 22456 34 9856 44 34572

5 64 15 1792 25 2607 35 7424 45 28480

6 448 16 4387 26 5888 36 57405 46 17728

7 116 17 616 27 5776 37 2816 47 4516

8 802 18 7120 28 14964 38 12224 48 122836

9 445 19 764 29 1744 39 10304 49 9009

10 1024 20 8256 30 28672 40 51328 50 41712 Define the multiplicative function h by

spn, n, nq “ÿ

d|n

d²τpdqhpn{dq pnP Nq (2.48) and letHpzq “ř8

n“1hpnqn^´z be the Dirichlet series of h.

Theorem 2.8.7 (Hampejs and T´oth [38, Th. 2.3]). For every ε ą0, ÿ

nďx

spn, n, nq “ x³

3 pHp3qplogx`2γ´1q `H¹p3qq `Opx<sup>2`θ`ε</sup>q, (2.49) where θ is the exponent in the Dirichlet divisor problem (1.1).

Chapter 3

Results for multiplicative functions of several variables

3.1 Counting r-tuples of positive integers with k-wise relatively prime components

Let r ě k ě 2 be fixed integers. The positive integers n₁, . . . , n_r are called k-wise relatively prime if anyk of them are relatively prime, that is pn_i1, . . . , n_ikq “1 for every 1ďi₁ ă ¨ ¨ ¨ ăi_k ďr. In particular, in the case k “2 the integers are pairwise relatively prime and fork “r they are mutually relatively prime.

Let S_r,k denote the set of r-tuples of positive integers with k-wise relatively prime components and let %_r,k stand for its characteristic function. What is the asymptotic density

d_r,k “ lim

xÑ8

1 x^r

ÿ

n1,...,nrďx

%_r,kpn₁, . . . , n_rq

of the setSr,k? Heuristically, the probability that a positive integer is divisible by a fixed prime pis 1{p, hence the probability that given r positive integers exactly j of them are divisible by p is

ˆr j

˙1 p^j

ˆ 1´ 1

p

˙r´j

and the probability that they are k-wise relatively prime is P_r,k “ź

p k´1

ÿ

j“0

ˆr j

˙1 p^j

ˆ 1´ 1

p

˙r´j

. (3.1)

Note that for every rěkě2,

c ź

pąr´1

ˆ

1´pr´1q² p²

˙

ďP_r,2 ďP_r,k ďP_r,r “ ź

p

ˆ 1´ 1

p^r

˙ ,

with some constant c ą 0 (depending on r), hence the infinite product (3.1) converges.

Some approximate values ofP_r,k are shown by Table 2.

Table 2. Approximate values of P_r,k for 2ďk ďr ď8 P_r,k k“2 k “3 k “4 k“5 k “6 k “7 k “8 r “2 0.607

r “3 0.286 0.831

r “4 0.114 0.584 0.923

r “5 0.040 0.357 0.768 0.964

r “6 0.013 0.195 0.576 0.873 0.982

r “7 0.004 0.097 0.394 0.734 0.930 0.991

r “8 0.001 0.045 0.247 0.573 0.837 0.962 0.995

If k “ r, then it is well known that d_r,r “ P_r,r “ 1{ζprq is the correct value of the corresponding asymptotic density. The casek“2 was treated by the author [95], proving by an inductive approach that

ÿ

n1,...,nrďx

%_r,2pn₁, . . . , n_rq “ d_r,2x^r`O`

x^r´1plogxq^r´1˘

, (3.2)

whered_r,2 “P_r,2.

In the case k “ 2, the asymptotic density was also deduced by Cai and Bach [14, Th. 3.3] using probabilistic arguments. J. Hu [43, 44] proved that d_r,k “ P_r,k for every rěkě2. In fact, by generalizing the method of [95] it was shown in [43] that

ÿ

n1,...,nrďx

%_r,kpn₁, . . . , n_rq “P_r,kx^r`O`

x^r´1plogxq^δr,k˘

, (3.3)

whereδ_r,k“max

!`_r´1

j

˘: 1ďj ďk´1 )

. Fork “2 the asymptotic formula (3.3) reduces to (3.2).

Referring to our paper [95], de Reyna and Heyman [29] considered modified pair-wise coprimality conditions and by using certain graph representations they obtained asymptotic formulas similar to (3.3). Probabilistic aspects of pairwise coprimality were investigated by Fern´andez and Fern´andez [35].

It is the goal of the present Section to use a method, which differs from all approaches mentioned above and which seems to be the most natural, to establish the asymptotic formula (3.3) with a better error term. More exactly, we take into account that the function%_r,kpn₁, . . . , n_rqis multiplicative, viewed as an arithmetic function ofr variables.

Therefore, its multiple Dirichlet series can be expressed as an Euler product and an explicit formula can be given for it. Then we use the convolution method to obtain the desired asymptotic formula by elementary arguments.

We prove the following results. Let e_jpx₁, . . . , x_rq denote the elementary symmetric polynomials in x₁, . . . , x_r of degreej (j ě0).

Theorem 3.1.1 (T´oth [111, Th. 2.1]). Let r ě k ě 2 and let s_i P C (1ď i ď r). If

<s_i ą1 (1ďiďr), then

8

ÿ

n1,...,nr“1

%_r,kpn₁, . . . , n_rq

n^s₁¹¨ ¨ ¨n^s_r^r “ζps₁q ¨ ¨ ¨ζps_rqD_r,kps₁, . . . , s_rq, where

D_r,kps₁, . . . , s_rq “ ź

p

˜ 1´

r

ÿ

j“k

p´1q^j´k

ˆj ´1 k´1

˙

e_jpp^´s1, . . . , p^´srq

¸

is absolutely convergent if <ps_i1 ` ¨ ¨ ¨ `s_ijq ą 1 for every 1 ď i₁ ă . . . ă i_j ď r with kďj ďr.

Theorem 3.1.2 (T´oth [111, Th. 2.2]). If rěkě2, then ÿ

n1,...,nrďx

%_r,kpn₁, . . . , n_rq “A_r,kx^r`OpR_r,kpxqq, where

A_r,k“ź

p

˜ 1´

r

ÿ

j“k

p´1q^j´k ˆr

j

˙ˆj´1 k´1

˙1 p^j

¸

(3.4) and

Rr,kpxq “

#x^r´1, if rěk ě3,

x^r´1plogxq^r´1, if rěk “2. (3.5) Fork ě3 the error term Rr,kpxq is better than in (3.3), obtained by J. Hu [43]. Note also that A_r,k “ P_r,k, given by (3.1), which follows by certain properties of the binomial coefficients.

3.2 The average value of the least common multiple of k positive integers

Consider the greatest common divisor pn₁, . . . , n_kqof n₁, . . . , n_kP N. It is easy to see that for any arithmetic functionf we have the identity

ÿ

n1,...,nkďx

fppn₁, . . . , n_kqq “ ÿ

dďx

pµ˚fqpdq Yx

d ]k

, (3.6)

which leads to asymptotic formulas for this sum. For example, if fpnq “ n and k ě 3, then we have

ÿ

n1,...,n_kďx

pn₁, . . . , n_kq “ ζpk´1q

ζpkq x^k`OpR_kpxqq,

where R₃pxq “ x²logx and R_kpxq “ x^k´1 for k ě 4. The case fpnq “ n, k “ 2 can be treated separately by writing

ÿ

m,nďx

pm, nq “ 2 ÿ

mďnďx

pm, nq ´ ÿ

nďx

n

“2ÿ

nďx

pµ˚idτqpnq ´ x²

2 `Opxq, giving, by using elementary arguments, the formula

ÿ

m,nďx

pm, nq “ x² ζp2q

ˆ

logx`2γ´1

2 ´ζp2q

2 ´ ζ¹p2q ζp2q

˙

`Opx<sup>1`θ`ε</sup>q, (3.7) valid for everyε ą0, where γ is Euler’s constant andθ is the exponent appearing in the Dirichlet divisor problem (1.1). Here (3.7) is equivalent to formula (2.20), concerning the gcd-sum function.

For the least common multiple of k positive integers there is no formula similar to (3.6). However, in the case k “2, the lcm of the integers m, nP N can be written using their gcd as rm, ns “ mn{pm, nq, which enables to establish the following asymptotic formula, valid for anyr PN:

ÿ

m,nďx

rm, ns^r “ ζpr`2q

ζp2q ¨ x^2pr`1q

pr`1q<sup>2</sup> `Opx^2r`1plogxq^2{3plog logxq^4{3q, which is a consequence of the result (1.4) of Walfisz onř

nďxφpnq.

The above and related results go back to the work of Ces`aro [18], Cohen[22], Diaconis and Erd˝os [31], Tanigawa and Zhai [92], Ikeda and Matsuoka [47], and others.

The result

ÿ

m,n,qďx

rm, n, qs^r „c_r x^3pr`1q

pr`1q³ pxÑ 8q,

valid for r P N, without any error term and with a computable constant c_r given in an implicit form, was obtained by J. L. Fern´andez and P. Fern´andez [34, Th. 3(b)].

Their proof is by an ingenious method based on the identityrm, n, qspm, nqpm, qqpn, qq “ mnqpm, n, qq (m, n, q P N) and using the dominated convergence theorem. As far as we know, there are no other asymptotic results in the literature for the sum

ÿ

n1,...,nkďx

fprn₁, . . . , n_ksq, (3.8)

in the casek ě3, wheref is an arithmetic function. It seems that the method of [34] can not be extended for k ě3, even in the case fpnq “n^r. Also, it is not possible to reduce the estimation of the sum (3.8) to sums of a single variable, like in (3.6).

In this Section we deduce an asymptotic formula with remainder term for the sum (3.8), wherek ě2 andf belongs to a large class of multiplicative arithmetic functions.

LetrPRbe a fixed number. LetA_r denote the class of complex valued multiplicative arithmetic functions satisfying the following properties: there exist real constants C₁, C₂ such that

|fppq ´p^r| ďC₁p^r´1{2 for every prime p, (i)

and

|fpp^νq| ďC₂p^νr for every prime power p^ν with νě2. (ii) Note that conditions (i) and (ii) imply that

|fpp^νq| ďC₃p^νr for every prime power p^ν with νě1, (iii) whereC₃ “maxpC₁`1, C₂q.

For example, the following functions belong to the classA_r: fpnq “n^r, σpnq^r, φpnq^r, σ^peqpnq^r (r P R), fpnq “ σ_rpnq “ ř

d|nd^r (r P R with r ě 1{2). Furthermore, if f is a bounded multiplicative function such thatfppq “1 for every prime p, then f PA₀. In particular, µ² PA₀.

Theorem 3.2.1 (Hilberdink and T´oth [42, Th. 2.1]). Let k ě 2 be a fixed integer and let f PA_r be a function, where r ą ´1 is real. Then for every εą0,

ÿ

n1,...,n_kďx

fprn₁, . . . , n_ksq “C_f,k x<sup>kpr`1q</sup> pr`1q^k `O

´

x<sup>kpr`1q´12</sup>minpr`1,1q`ε¯

, (3.9)

and

ÿ

n1,...,nkďx

fprn₁, . . . , n_ksq

pn1¨ ¨ ¨nkq^r “C_f,kx^k`O

´

x^k´12minpr`1,1q`ε¯

, (3.10)

where

C_f,k “ź

p

ˆ 1´ 1

p

˙k 8

ÿ

ν1,...,νk“0

fpp^{maxpν1,...,νkq}q p^{pr`1qpν1`¨¨¨`νkq}.

Formula (3.9) shows that the average order offprn₁, . . . , n_ksqisC_f,kpn₁¨ ¨ ¨n_kq^r, in the sense that

ÿ

n1,...,nkďx

fprn₁, . . . , n_ksq „ ÿ

n1,...,nkďx

C_f,kpn₁¨ ¨ ¨n_kq^r pxÑ 8q. From (3.10) we deduce that

xÑ8lim 1 x^k

ÿ

n1,...,nkďx

fprn1, . . . , nksq

pn₁¨ ¨ ¨n_kq^r “Cf,k,

representing the mean value of the functionfprn₁, . . . , n_ksq{pn₁¨ ¨ ¨n_kq^r.

Theorem 3.2.2 (Hilberdink and T´oth [42, Th. 2.2]). Let k ě 2 be a fixed integer Note that our method does not work in the case r “ ´1, that is for the sum

S_kpxq:“ ÿ

n1,...,nkďx

1 rn₁, . . . , n_ks.

By using different arguments, we proved (unpublished) that S_kpxq — plogxq^2k´1 for every kě2, this sum being related to ř

nďxτpnq^k{n. However, we are not able to obtain an asymptotic formula for S_kpxq.

Among other special cases of the above results we consider here only the function σPA₁.

Corollary 3.2.4 (Hilberdink and T´oth [42, Cor. 3]). Let k ě 2. Then for every εą0,

ÿ

n1,...,nkďx

σprn₁, . . . , n_ksq “C_σ,kx^2k 2^k `O`

x^2k´1{2`ε˘ , and

ÿ

n1,...,n_kďx

σprn₁, . . . , n_ksq

n₁¨ ¨ ¨n_k “Cσ,kx^k`O`

x^k´1{2`ε˘ , where

C_σ,k “ź

p

ˆ 1´ 1

p

˙k 8

ÿ

ν1,...,νk“0

σpp^{maxpν1,...,νkq}q p^{2pν1`¨¨¨`νkq} . In particular,

Cσ,2 “ζp3qζp4qź

p

ˆ 1` 1

p² ´ 2 p³ ´ 2

p⁵ ` 2 p⁶

˙ .

In document Asymptotic Properties of Multiplicative Arithmetic Functions of One and Several Variables DSc dissertation (Pldal 39-47)