On the average number of direct factors of finite abelian groups (II)

XC.1 (1999)

On the average number of direct factors of finite abelian groups (II)

by

Wenguang Zhai (Jinan)

1. Introduction. Let a : n → a(n) be as usual the arithmetic function which counts the number of finite abelian groups of given order n and define t = a ∗ a and ω = a ∗ a ∗ a. We shall be concerned with obtaining estimates for the sums T (x) = P

n≤x

t(n) and W (x) = P

n≤x

ω(n).

The asymptotic behaviour of T (x) was first studied by Cohen [1], who derived that

(1.1) T (x) = c

x(log x + 2γ − 1) + c

x + ∆

(x) with ∆

(x)  √

x log x. Kr¨atzel [6] improved this result to (1.2) ∆

(x) = c

√

x((log x)/2 + 2γ − 1) + c

√

x + ∆

(x)

with ∆

(x)  x

log

x. The exponent 5/12 was improved to 83/201, 45/109, 9/22, 3/8, 7/19, 4/11 by Menzer [8], Menzer and Seibold [10], Men- zer [9], Yu [14], Liu [7], Zhai and Cao [15], respectively. It should be men- tioned that recently J. Wu [13] has obtained a better exponent 47/131.

H. Menzer [9] studied the asymptotic behaviour of W (x). He proved that (1.3) W (x) = xP

(log x) + √

xP

(log x) + O(x

log

x), where P

(j = 1, 2) denotes a polynomial of degree 2.

The aim of this short note is to further improve Menzer’s result by a different approach giving

Theorem 1. We have the asymptotic formula (1.4) W (x) = xP

(log x) + √

xP

(log x) + O(x

).

1991 Mathematics Subject Classification: Primary 11N37.

This work is supported by National Natural Science Foundation of China (Grant No.

19801021).

[69]

Following H. Menzer [9], we only need to study the asymptotic behaviour of the divisor function

d(1, 1, 1, 2, 2, 2; n) = X

n1n2n3m²₁m²₂m²₃=n

1.

Let ∆(1, 1, 1, 2, 2, 2; x) denote the error term of the summation function D(1, 1, 1, 2, 2, 2; x) = X

n≤x

d(1, 1, 1, 2, 2, 2; n).

Then Theorem 1 follows from Theorem 2. We have

(1.5) ∆(1, 1, 1, 2, 2, 2; x) = O(x

).

The key of the proof is contained in Lemma 8 of Section 2, which con- nects the problem with the well-known Piltz divisor problem. So the cor- responding exponential sums are bilinear forms which can be estimated by the well-known double large sieve inequality due to Bombieri and Iwaniec (Proposition 1 of Fouvry and Iwaniec [2]; see Lemma 1 below). A detailed proof of Theorem 2 is given in Section 3.

Notations. e(t) = exp(2πit). n ∼ N means C

N < n < C

N for some absolute constants C

and C

. ε is a sufficiently small number which may be different at each occurrence. ∆

(t) always denotes the error term of the Piltz divisor problem. We use notation SC(Σ) to denote the summation conditions of the sum Σ if these conditions are complicated. For example, instead of

F (x) = X

a≤n≤x

f (n) we write

F (x) = X

f (n), SC(Σ) : a ≤ n ≤ x.

2. Some preliminary lemmas. We need the following lemmas.

Lemma 1. Let X and Y be two finite sets of real numbers, X ⊂ [−X, X], Y ⊂ [−Y, Y ]. Then for any complex functions u(x) and v(y) we have

   X

x∈X

X

y∈Y

u(x)v(y)e(xy)   

≤ 20(1 + XY ) X

x,x⁰∈X

|x−x⁰|≤Y⁻¹

|u(x)u(x

)| X

y,y⁰∈Y

|y−y⁰|≤X⁻¹

|v(y)v(y

)|.

P r o o f. This is Proposition 1 of Fouvry and Iwaniec [2].

Lemma 2. Suppose that 0 < a < b ≤ 2a and R is an open convex set in C containing the real segment [a, b]. Suppose further that f (z) is analytic on R, f (x) is real for real x in R, |f

(z)| ≤ M for z ∈ R, and there is a constant k > 0 such that f

(x) ≤ −kM for all real x in R. Let α = f

(b), β = f

(a) and define x

for each integer v in the range α < v < β by f

(x

) = v. Then

X

a<n≤b

e(f (n)) = e(−1/8) X

α<v≤β

|f

(x

)|

e(f (x

) − vx

) + O(M

+ log(2 + M (b − a))).

P r o o f. This is Lemma 6 of Heath-Brown [3].

Lemma 3. Let αβ 6= 0, ∆ > 0, M ≥ 1 and N ≥ 1. Let A(M, N ; ∆) be the number of quadruples (m

, m

, n

, n

) such that

 m

m



−

 n

n



    < ∆ with M ≤ m

, m

≤ 2M and N ≤ n

, n

≤ 2N . Then

A(M, N ; ∆)  M N log 2M N + ∆M

N

. P r o o f. This is Lemma 1 of Fouvry and Iwaniec [2].

Lemma 4. Let 0 < L ≤ N < M ≤ cL and a

be complex numbers such that |a

| ≤ 1. Then

X

N <n≤M

a

= 1 2π

cL

\

−cL

 X

L<l≤cL

a

l



(M

− N

)t

dt + O(log(2 + L)).

P r o o f. This is essentially Lemma 6 of Fouvry and Iwaniec [2].

Lemma 5. For 1  Y  x

we have

∆

(x) = x

√ 3 π X

1≤n≤Y

d

(n)

n

cos(6(nx)

) + O(x

Y

).

P r o o f. See Chapter 3 of Ivi´c [4] or p. 80 of Min [11].

Lemma 6. Let M > 0, N > 0, u

> 0, v

> 0, A

> 0, B

> 0 (1 ≤ m ≤ M, 1 ≤ n ≤ N ), and let Q

and Q

be given non-negative numbers, Q

≤ Q

. Then there is a q such that Q

≤ q ≤ Q

and

X

A

q

+ X

B

q

 X

X

(A

B

)

+ X

A

Q

+ X

B

Q

.

P r o o f. See Srinivasan [12].

Lemma 7. Suppose M ≥ 2, N ≥ 2, z ≥ 2 are positive numbers, α and β are real non-integer constants and F = zM

N

, a(m)  d

(m), b(n)  d

(n). Then

X

m∼M

a(m) X

n∼N

b(n)e(zm

n

)

 (F M N )

 1 + M

F



 1 + N

F



log

F M N.

P r o o f. This can be easily derived from Theorem 1 of Fouvry and Iwa- niec [2] if we notice P

n≤x

d

(n)  x log

x.

Lemma 8. We have (2.1) ∆(1, 1, 1, 2, 2, 2; x)

= X

m≤x^1/3

d

(m)∆

 x m



+ X

m≤x^1/3

d

(m)∆

r x m



+ O(x

).

P r o o f. This lemma plays an important role in our proof and the same idea has been used in Zhai and Cao [15].

We only sketch the proof since it is elementary and direct. We leave the details to the reader.

We begin with

(2.2) D(1, 1, 1, 2, 2, 2; x)

= X

n≤x

d(1, 1, 1, 2, 2, 2; n)

= X

n1n2n3m²₁m²₂m²₃≤x

1 = X

m²n≤x

d

(n)d

(m)

= X

n≤x^1/3

d

(n)D

r x n



+ X

m≤x^1/3

D

 x m



− D

(x

)

= Σ

+ Σ

− D

(x

), say, where D

(x) = P

n≤x

d

(n). We use the following abelian partial sum- mation formula:

(2.3) X

n≤u

d

(n)f (n) = D

(u)f (u) −

u

\

1

D

(t)f

(t) dt to Σ

and Σ

. For D

(u), we use

(2.4) D

(u) = d

u log

u + d

u log u + d

u + ∆

(u)

with ∆

(u)  u

and (2.5)

T

\

1

∆

(u) du  T

(T ≥ 2).

Formula (2.4) is in Chapter 13 of Ivi´c [4]. Formula (2.5) can be easily derived from Lemma 5.

After some calculations, we can get

D(1, 1, 1, 2, 2, 2; x) = main terms + X

m≤x^1/3

d

(m)∆

 x m

 (2.6)

+ X

n≤x^1/3

d

(n)∆

r x n



+ O(x

),

whence our lemma follows.

3. Proof of Theorem 2. In order to prove Theorem 2, we only need to estimate the two sums in Lemma 8. We first prove the following

Proposition 1. We have the estimate X

m≤x^1/3

d

(m)∆

 x m



= O(x

).

P r o o f. We only need to show that

(3.1) S(M ) = X

m∼M

d

(m)∆

 x m



 x

for any fixed 1  M  x

.

Case 1: M  x

. In this case (3.1) follows from Kolesnik’s well-known estimate ∆

(u)  u

(see Kolesnik [5]).

Case 2: x

 M  x

. Suppose 1  Y  x is a parameter to be determined. By Lemma 5 we get

S(M )  x

X

m∼M

d

(m) m

X

n≤Y

d

(n) n

e

 3(nx)

m

  

 + x

(Y M )

+ x

(3.2)

 x

|S(M, N )| log x + x

(Y M )

+ x

for some 1  N  Y , where

(3.3) S(M, N ) = X

m∼M

d

(m) m

X

n∼N

d

(n) n

e

 3(nx)

m



.

We choose Y = xM

, thus x

(Y M )

 (xM )

x

 x

.

Let a(m) = d

(m)(M/m)

, b(n) = d

(n)(N/n)

, F = (xN )

M

, and

T (M, N ) = X

m∼M

a(m) X

n∼N

b(n)e

 3(nx)

m

 . Obviously

(3.4) S(M, N )  x

(M N )

|T (M, N )|.

Since M  x

, it is easy to check that F  M . If N  x

/M , then F  N . By Lemma 7 we can get

T (M, N )  (F M N )

log

x, which combined with (3.4) gives

(3.5) S(M, N )  x

M

log

x  x

. Now suppose N  x

/M . Using the expression d

(n) = P

n=uvw

1, we find that T (M, N ) can be divided into O(log

x) sums of the form

T (M, U, V, W ) = X

m∼M

a(m) X

(u,v,w)

e

 3(uvwx)

m

 , where

SC  X

(u,v,w)



: N ≤ uvw ≤ 2N, U < u ≤ 2U, V < v ≤ 2V, W < w ≤ 2W, u ≤ v ≤ w, U V W ∼ N.

It follows that W  N

. If W  N

, then by the exponent pair (1/2, 1/2) we have

T (M, U, V, W ) log

x  M U V F

+ M N/F (3.6)

 F

M N

+ M N/F, whose contribution to S(M, N ) is

 F

M N

log

x + M log

x  x

N

log

x + M log

x

 x

M

log

x + M log

x  x

log

x,

where we used the assumptions N  x

/M and M  x

.

Later we always suppose N

 W  N

, namely, N

 U V 

N

. Let a = max(N/(uv), W, v), b = min(2W, 2N/(uv)). Using Lemma 2

to the variable w we have

(3.7) X

a≤w≤b

e

 3(uvwx)

m



= c

X

r

(xuv)

m

r

e

 2(xuv)

mr

 + O



log x + W F

 ,

where

SC  X

r



: B = (xuv)

(mb)

≤ r ≤ (xuv)

(ma)

= A, r ∼ R = F/W.

Using Lemma 4 to the variable r we find that

(3.8) X

a≤w≤b

e

 3(uvwx)

m



= c

2π

100F/W

\

F/(100W )

 X

F/(100W )≤r≤100F/W

(xuv)

m

r

e

 2(xuv)

mr



× A

− B

t dt + O(log x + W F

log x).

Thus we get (3.9) T (M, U, V, W )

 W

F

X

m∼M

X

r∼R

d

(m)    

X

(u,v)

c

(u)c

(v)e

 2(xuv)

mr

   

+ M N

log x + M N F

log x for some c

(u)  1 and some c

(v)  1, where

SC  X

(u,v)



: U ≤ u < 2U, V ≤ v < 2V, u ≤ v.

Let T

denote the exponential sum in the right side of (3.9). By Lemma 1 we get

(3.10) |T

|

 F AB,

where

A = X

∗

d

(m)d

(m

), B = X

∗∗

1

with

SC  X

∗



: |m

r

− m

r

|  (xU V )

, m ∼ M, r ∼ R, SC  X

∗∗



: |u

v

− u

v

|  M R

x

, u ∼ U, v ∼ V.

By Lemma 3 we get

(3.11) Ax

 M R + (xU V )

M

R

and

B  U V log x + M R

x

(U V )

(3.12)

 U V log x + (U V )

/F  U V log x, where we used the fact that F  N

.

Combining (3.9)–(3.12) we get

T (M, U, V, W )x

 F

(U V )

M

x

W

(3.13)

+ (F M N )

+ M N

+ M N F

 (F M N )

+ x

F

M

N

+ M N

+ M N F

,

whose contribution to S(M, N ) is

 F x

N



(F M N )

+ x

F

M

N

+ M N

+ M N F



 (x

M

+ x

N

+ x

M

+ x

M

N

)x

 (x

M

+ x

M

+ x

M

)x

 x

, if we use the assumptions x

M

 N  xM

and M  x

.

Combining the above we see that (3.1) holds in Case 2.

Case 3: x

 M  x

. We begin with (3.2). Using Lemma 7 directly to bound T (M, N ) we can get

S(M )x

 x

M

+ (xM )

+ (xY )

M

(3.14)

+ x

M

Y

+ x

(M Y )

. Choosing a best Y ∈ [1, x

] via Lemma 6 we get

(3.15) S(M )x

 x

M

+ x

M

 x

. This completes the proof of Proposition 1.

The second sum in Lemma 8 is handled in

Proposition 2. We have the estimate X

m≤x^1/3

d

(m)∆

r x m



= O(x

).

P r o o f. It suffices to prove (3.16) S

(M ) = X

m∼M

d

(m)∆

r x m



 x

for 1  M  x

.

For M  x

, we use the trivial bound ∆

(u)  u

. For x

 M  x

, the proof of (3.16) is the same as that of Case 3 of Proposition 1.

This completes the proof of Proposition 2.

Theorem 2 immediately follows from Lemma 8 and the two propositions.

Acknowledgements. The author thanks the referee for valuable sug- gestions and for indicating Wu’s paper [13].

References

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Department of Mathematics Shandong Normal University Jinan, 250014, Shandong P.R. China

E-mail: wgzhai@jn-public.sd.cninfo.net

Received on 15.9.1998

and in revised form on 19.1.1999 (3462)