Proofs for Section 2.8.2

We need the next general result regarding the subgroups of the group GˆZq, where pG,`q is an arbitrary finite Abelian group. For a subgroup H of G (notation H ď G) consider the congruence relation%_H on Gdefined for x, x¹ PG byx%_Hx¹ if x´x¹ PH.

Lemma 4.8.1. For a finite Abelian group pG,`q and q PN let

IG,q :“ tpH, α, dq:H ďG, αPSH, d|q and pq{dqαPHu,

where S_H is a complete system of representants of the equivalence classes determined by

%_H. For pH, α, dq PI_G,q define

V_H,α,d :“ tpkα`β, kdq: 0ďk ďq{d´1, β PHu.

Then V_H,α,d is a subgroup of order pq{dq#H of GˆZq and the map pH, α, dq ÞÑ V_H,α,d is a bijection between the set I_G,q and the set of subgroups of GˆZq.

Proof. LetV be a subgroup ofGˆZq. Consider the natural projectionπ₂ :GˆZqÑZq

given by π₂px, yq “y. Thenπ₂pVqis a subgroup of Zq and there is a unique divisor d of q such that π2pVq “ xdy :“ tkd: 0ďk ďq{d´1u. Let αP Gsuch that pα, dq PV.

Furthermore, consider the natural inclusion ι₁ :G ÑGˆZq given by ι₁pxq “ px,0q. Then ι^´1₁ pVq “H is a subgroup of G. We show that V “ tpkα`β, kdq:k PZ, β PHu.

Indeed, for every k P Z and β P H, pkα`β, kdq “ kpα, dq ` pβ,0q P V. On the other hand, for every pu, vq PV one has v P π₂pVq and hence there is k P Z such that v “kd.

We obtainpu´kα,0q “ pu, vq ´kpα, dq PV, thus β :“u´kαPι^´1₁ pVq “ H.

Here a necessary condition is thatpq{dqα PH (obtained fork “q{d, β“0). Clearly, if this is verified, then for the above representation of V it is enough to take the values 0ďk ďq{d´1.

Conversely, every pH, α, dq P IG,q generates a subgroup VH,α,d of order pq{dq#H of GˆZq. Furthermore, for fixed H ďG and d | q we have V_H,α,d “H_H,α¹_,d if and only if α%_Hα¹. This completes the proof.

In the caseG“Z^m (and with q“n) Lemma 4.8.1 can be stated as follows:

Lemma 4.8.2. For every m, nP N let

I_m,n :“ tpa, b, sq PN²ˆNY t0u:a |m, b|n,0ďsďa´1 and a| pn{bqsu and for pa, b, sq PIm,n define

V_a,b,s :“ xpa,0q,pb, sqy (4.32)

“ tpia`js, jbq: 0ďiďm{a´1,0ďj ďn{b´1u.

Then V_a,b,s is a subgroup of order ^mn_ab of Zm ˆZn and the map pa, b, sq ÞÑ V_a,b,s is a bijection between the set I_m,n and the set of subgroups of ZmˆZn.

Note thata| pn{bqsholds if and only ifa{gcdpa, n{bq |s. That is, for sPI_m,n we have s“ at

A, 0ďtďA´1, (4.33)

where A “ gcdpa, n{bq, notation given in Theorem 2.8.5. This quickly leads to formula (2.44) regarding the number spm, nq of subgroups of ZmˆZn, namely

spm, nq “ ÿ

a|m,b|n

ÿ

0ďtďA´1

1“ ÿ

a|m,b|n

gcdpa, bq.

deduced by different arguments.

Proof of Theorem 2.8.5. Apply Lemma 4.8.1 for G “ Zm ˆZn and with q “ r. For the subgroups V “ V_a,b,s given by Lemma 4.8.2 a complete system of representants of the equivalence classes determined by %_V is S_a,b “ t0,1, . . . , a´1u ˆ t0,1, . . . , b´1u.

Indeed, the elements of S_a,b are pairwise incongruent with respect to V, and for every px, yq P ZmˆZn there is a unique px¹, y¹q P S_a,b such that px, yq ´ px¹, y¹q PV. Namely, let

px₁, y₁q “ px, yq ´ty{bups, bq, px¹, y¹q “ px₁, y₁q ´tx₁{aupa,0q. We obtain that the subgroups of ZmˆZnˆZr are of the form

U “U_H,α,c “ tpkα`β, kcq: 0ďk ďr{c´1, β PVu, wherec|r and α“ pu, vq PS_a,b such that pr{cqαPV.

Now using (4.32) we deduce

U “U_a,b,s,α,c

“ tpia`js`ku, jb`kv, kcq: 0ďiďn{a´1,0ďj ďn{b´1,0ďk ďn{c´1u, where (4.33) holds and pr{cqpu, vq P V. From the latter condition we deduce that there are i₀, j₀ such that

pr{cqu“i₀a`j₀s, pr{cqv “j₀b. (4.34)

The second condition of (4.34) holds if b| pr{cqv, that is b{gcdpb, r{cq |v. Let v “ bv₁

B , 0ďv₁ ďB´1, (4.35)

where B “ gcdpb, r{cq. Also, j₀ “ rv{pbcq and inserting this into the first equation of (4.34) we obtain pr{cqu ” rvs{pbcq (mod a). This linear congruence in u has a solution u₀ if and only if

gcdpa, r{cq | rvs

bc (4.36)

and its all solutions areu“u₀`az{C with 0ďz ďC´1 with C “gcdpa, r{cq.

Substituting (4.35) and (4.33) into (4.36) we obtain gcdpa, r{cq | rab

gcdpab, nqgcdpbc, rqv₁t, that is

gcdpab, nqgcdpac, rqgcdpbc, rq |abcrv₁t, equivalent to

gcdpab, nqgcdpac, rqgcdpbc, rq

gcdpabcr,gcdpab, nqgcdpac, rqgcdpbc, rqq |v1t, and to

X |v₁t, (4.37)

where X is defined in the statement of Theorem 2.8.5. Note that X | B (indeed, A | a, C| pr{cqand the property follows from X “B{gcdppa{Aqpr{cq{Cq, B).

Lett be fixed. We obtain from (4.37) thatv₁ is of the formv₁ “Xw{gcdpt, Xq, where 0 ď w ď Bgcdpt, Xq{X ´1. Also, from (4.35), v “ bXw{Bgcdpt, Xq. Collecting the conditions ona, b, c, t, w, z in terms ofA, B, C, X finishes the proof.

Proof of Theorem 2.8.6. According to Theorem 2.8.5, the number of subgroups of Γ is spm, n, rq “ ÿ

a|m,b|n,c|r

ÿ

0ďtďA´1

ÿ

0ďwďBgcdpt,Xq{X´1

ÿ

0ďzďC´1

1

“ ÿ

a|m,b|n,c|r

C ÿ

0ďtďA´1

B

X gcdpt, Xq “ ÿ

a|m,b|n,c|r

BC X

ÿ

1ďtďA

gcdpt, Xq.

Here X |A (similar to X |B shown above), hence the inner sum is pA{XqPpXq and we obtain formula (2.47).

Proof of Theorem 2.8.7. Letspnq “spn, n, nq. According to (2.48), the function spnq can

gcdpapr{cq, ABCqPpXq{X ď ÿ

a,b,c|n From the Euler product formula

Hpzq “ź

Furthermore, by partial summation we obtain from (1.1) that ÿ and inserting (4.41) we get

ÿ

. This gives the asymptotic formula (2.49).

Chapter 5

Proofs of the results of Chapter 3

5.1 Proofs of the results of Section 3.1

Proof of Theorem 3.1.1. We use the polynomial identity (proved in our paper [111])

k´1 right hand side shows how it can be written as a polynomial of the elementary symmetric polynomials.

“ζps1q ¨ ¨ ¨ζpsrq

The function ψ_r,k is also multiplicative, symmetric in the variables and for any prime powers p^ν1, . . . , p^νr,

From (5.2) we deduce ÿ

where the first sum is overu₁, . . . , u_r P t0,1usuch that at least oneu_i is 0. Let u₁, . . . , u_r be fixed and assume that u_r “0. Since px{d_iq^ui ďx{d_i for every i, we have

A:“x^u1`¨¨¨`ur ÿ

d1,...,drďx

|ψ_r,kpd₁, . . . , d_rq|

d^u₁¹¨ ¨ ¨d^u_r^r ďx^r´1 ÿ

d1,...,drďx

|ψ_r,kpd₁, . . . , d_rq|

d₁¨ ¨ ¨d_r´1 Assume that k ě3. Then

Aďx^r´1

8

ÿ

d1,...,dr“1

|ψ_r,kpd₁, . . . , d_rq|

d₁¨ ¨ ¨d_r´1 !x^r´1,

since the series Dr,kp1, . . . ,1,0q is absolutely convergent for k ě3 by Theorem 3.1.1. We obtain that

Q_r,kpxq !x^r´1 pk ě3q. (5.5)

Ifk “2, then

Aďx^r´1ź

pďx 8

ÿ

ν1,...,νr“0

|ψ_r,2pp^ν1, . . . , p^νrq|

p^{ν1`¨¨¨`νr´1} (5.6)

“x^r´1ź

pďx

ˆ

1`r´1 p ` c₂

p² ` ¨ ¨ ¨ ` c_r´1 p^r´1

˙ ,

by (5.3), where c2, . . . , cr´1 are certain positive integers, using also that we have p in the denominator if and only if ν_r “ 1 and exactly one of ν₁, . . . , ν_r´1 is 1, the rest being 0, which occursr´1 times. We deduce that

A!x^r´1ź

pďx

ˆ 1`1

p

˙r´1

!x^r´1plogxq^r´1 by Mertens’ formula. This shows that

Q_r,2pxq !x^r´1plogxq^r´1. (5.7) Furthermore, for the main term of (5.4) we have

ÿ

d1,...,drďx

ψr,kpd1, . . . , drq d₁¨ ¨ ¨d_r

“

8

ÿ

d1,...,dr“1

ψr,kpd1, . . . , drq d₁¨ ¨ ¨d_r ´

ÿ

H‰IĎt1,...,ru

ÿ

diąx, iPI djďx, jRI

ψr,kpd1, . . . , drq

d₁¨ ¨ ¨d_r , (5.8) where the series is convergent by Theorem 3.1.1 and its sum isD_r,kp1, . . . ,1q “A_r,k, given by (3.4).

LetI be fixed and assume thatI “ t1,2, . . . , tu, that isd₁, . . . , d_tąxandd_t`1, . . . , d_r ď x, wheretě1. We estimate the sum

B :“ ÿ

d1,...,dtąx dt`1,...,drďx

|ψ_r,kpd₁, . . . , d_rq|

d₁¨ ¨ ¨d_r by distinguishing the following cases:

Case i) kě3,t ě1:

B ă 1 x

8

ÿ

d1,...,dr“1

|ψ_r,kpd₁, . . . , d_rq|

d2¨ ¨ ¨dr

! 1 x, since the series is convergent by Theorem 3.1.1.

Case ii)k “2,t ě3: if 0 ăεă1{2, then B “

ÿ

d1,...,dtąx dt`1,...,drďx

|ψ_r,2pd₁, . . . , d_rq|d^ε´1{2₁ ¨ ¨ ¨d^ε´1{2_t d^{1{2`ε</sup><sub>1</sub> ¨ ¨ ¨d<sup>1{2`ε}_t dt`1¨ ¨ ¨dr

ăx^tpε´1{2q

8

ÿ

d1,...,dr“1

|ψ_r,2pd₁, . . . , d_rq|

d^{1{2`ε</sup><sub>1</sub> ¨ ¨ ¨d<sup>1{2`ε}_t d_t`1¨ ¨ ¨d_r !x^tpε´1{2q,

since the series is convergent (fortě1). Using thattpε´1{2q ă ´1 for 0ăεă pt´2q{p2tq, here we needt ě3, we obtain B ! ¹_x.

Case iii) k “ 2, t “ 1: Let d1 ą x, d2, . . . , dr ď x and consider a prime p. If p | di

for an i P t2, . . . , ru, then p ď x. If p | d₁ and p ą x, then p - d_i for every i P t2, . . . , ru and ψ_r,2pd₁, . . . , d_rq “0 by its definition (5.3). Hence it is enough to consider the primes pďx. We deduce

B ă 1 x

ÿ

d1ąx d2,...,drďx

|ψ_r,2pd₁, . . . , d_rq|

d₂¨ ¨ ¨d_r

ď 1 x

ź

pďx 8

ÿ

ν1,...,νr“0

|ψ_r,2pp^ν1, . . . , p^νrq|

p^{ν2`¨¨¨`νr} ! 1

xplogxq^r´1, similar to the estimate of (5.6).

Case iv) k “2, t“2: We split the sum B into two sums, namely

B “ ÿ

d1ąx,d2ąx d3,...,drďx

|ψ_r,2pd₁, . . . , d_rq|

d₁¨ ¨ ¨d_r

“ ÿ

d1ąx^3{2,d2ąx d3,...,drďx

|ψ_r,2pd₁, . . . , d_rq|

d₁¨ ¨ ¨d_r ` ÿ

x^3{2ěd1ąx,d2ąx d3,...,drďx

|ψ_r,2pd₁, . . . , d_rq|

d₁¨ ¨ ¨d_r “:B₁`B₂, say, where

B₁ “

ÿ

d1ąx^3{2,d2ąx d3,...,drďx

|ψ_r,2pd₁, . . . , d_rq|

d^1{3₁ d2¨ ¨ ¨dr

1 d^2{3₁

ă 1 x

8

ÿ

d1,...,dr“1

|ψr,2pd1, . . . , drq|

d^1{3₁ d₂¨ ¨ ¨d_r ! 1 x, since the series is convergent. Furthermore,

B₂ ă 1 x

ÿ

x^3{2ěd1,d2ąx d3,...,drďx

|ψ_r,2pd₁, . . . , d_rq|

d₁d₃¨ ¨ ¨d_r ,

where d₁ ď x^3{2, d₂ ą x, d₃, . . . , d_r ď x. Consider a prime p. If p | d_i for an i P t1,3, . . . , ru, then pďx^3{2. Ifp|d2 and pąx^3{2, thenp-di for everyiP t1,3, . . . , ruand ψ_r,2pd₁, . . . , d_rq “ 0 by its definition. Hence it is enough to consider the primes pďx^3{2. We deduce, cf. the estimate of (5.6),

B₂ ă 1 x

ź

pďx^3{2 8

ÿ

ν1,...,νr“0

|ψ_r,2pp^ν1, . . . , p^νrq|

p^{ν1`ν3`¨¨¨`νr} ! 1

xplogx^3{2q^r´1 ! 1

xplogxq^r´1.

Hence, given any t ě 1, we have B ! ¹_x for k ě 3 and B ! ¹_xplogxq^r´1 for k “ 2.

Therefore, by (5.8),

ÿ

d1,...,drďx

ψ_r,kpd₁, . . . , d_rq

d₁¨ ¨ ¨d_r “A_r,k`OpR_r,kpxqq (5.9) with the notation (3.4) and (3.5).

The proof is complete by putting together (5.4), (5.5), (5.7) and (5.9).

In document Asymptotic Properties of Multiplicative Arithmetic Functions of One and Several Variables DSc dissertation (Pldal 84-92)