1. Introduction. Let t(G) denote the number of unitary factors of G and

LXXXV.4 (1998)

On the average number of unitary factors of finite abelian groups

by

Wenguang Zhai (Jinan) and Xiaodong Cao (Beijing)

1. Introduction. Let t(G) denote the number of unitary factors of G and

T (x) = X t(G),

where the summation is taken over all abelian groups of order not exceeding x. It was first proved by Cohen [2] that

(1.1) T (x) = c

x(log x + 2γ − 1) + c

x + E

(x) with E

(x)  √

x. Kr¨atzel [3] proved that E

(x) = c

√

x + E

(x) with E

(x)  x

log

x.

Let θ denote the smallest α such that

(1.2) E

(x)  x

.

Then the exponents θ ≤ 31/82, 3/8, 77/208 were obtained by H. Menzer [7], P. G. Schmidt [8], H. Q. Liu [6], respectively.

The aim of this paper is to further improve the exponent 77/208. We have the following Theorem 1.

Theorem 1. θ ≤ 9/25.

Following Kr¨atzel [3], we only need to study the asymptotic behavior of the divisor function d(1, 1, 2; n) which is defined by

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

n1n2n²₃=n

1.

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

n≤x

d(1, 1, 2; n).

1991 Mathematics Subject Classification: Primary 11N37.

[293]

We then have

Theorem 2. ∆(1, 1, 2; x) = O(x

).

Theorem 1 immediately follows from Theorem 2.

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 Dirichlet divisor problem; ψ(t) = t − [t] − 1/2.

2. A non-symmetric expression of ∆(1, 1, 2; x). In this paper we shall use a non-symmetric expression of ∆(1, 1, 2; x) instead of its symmetric expression used in previous papers. This is the following lemma.

Basic Lemma. We have

∆(1, 1, 2; x) = X

m≤x^1/3

∆

 x m



+ O(x

log x).

P r o o f. We only sketch the proof since it is elementary and direct. We begin with

D(1, 1, 2; x) = X

n≤x

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

n₁n₂n²₃≤x

1 = X

m²n≤x

d(n)

= X

n≤x^1/3

d(n)[ p

x/n] + X

n≤x^1/3

D

 x m



− [x

]D(x

), where D(x) = P

n≤x

d(n). We apply the well-known abelian partial sum- mation formula

X

n≤u

d(n)f (n) = D(u)f (u) −

u

\

1

D(t)f

(t) dt

to the first sum in the above expression and the Euler–Maclaurin summation formula

X

n≤x

f (n) =

u

\

1

f (t) dt + f (1)

2 − ψ(u)f (u) +

u

\

1

D(t)f

(t) dt to the second sum, and combine

D(u) = u log u + (2γ − 1)u + ∆(u) with ∆(u)  u

to get

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

m≤x^1/3

∆

 x m



− X

n≤x^1/3

d(n)ψ( p

x/n) + O(x

)

= main terms + X

m≤x^1/3

∆

 x m



+ O(x

log x),

whence our lemma follows.

3. Some preliminary lemmas

Lemma 1. Suppose 0 < c

λ ≤ |f

(x)| ≤ c

λ and |f

(x)| ∼ λN

for N ≤ n ≤ cN . Then

X

a<n≤cN

e(f (n))  (λN )

+ λ

.

Lemma 2 (see [4]). Suppose f (x) and g(x) are algebraic functions in [a, b]

and

|f

(x)| ∼ R

, |f

(x)|  (RU )

,

|g(x)|  G, |g

(x)|  GU

, U, U

≥ 1.

Then X

a<n≤b

g(n)e(f (n)) = X

α<u≤β

b

g(n

)

p |f

(n

)| e(f (n

) − un

+ 1/8)

+ O(G log(β − α + 2) + G(b − a + R)(U

+ U

)) + O



G min √

R, 1 hαi



+ G min √

R, 1 hβi



, where [α, β] is the image of [a, b] under the mapping y = f

(x), n

is the solution of the equation f

(x) = u,

b

=

 1 for α < u < β,

1/2 for u = α ∈ Z or u = β ∈ Z;

and the function hti is defined as follows:

hti =

 ktk if t is not an integer , β − α otherwise,

where ktk = min

{|t − n|}.

Lemma 3 (see [5]). Let H ≥ 1, X ≥ 1, Y ≥ 1000; let α, β and γ be real numbers such that αγ(γ − 1)(β − 1) 6= 0, and A > C(α, β, γ) > 0, f (h, x, y) = Ah

x

y

. Define

S(H, X, Y ) = X

(h,x,y)∈D

c

(h, x)c

(y)e(f (h, x, y)),

where D is a region contained in the rectangle

{(h, x, y) | h ∼ H, x ∼ X, y ∼ Y }

such that for any fixed pair (h

, x

), the intersection D∩{(h

, x

, y) | y ∼ Y } has at most O(1) segments. Also, suppose

|c

(h, x)| ≤ 1, |c

(y)| ≤ 1, F = AH

X

Y

 Y.

Then

L

S(H, X, Y )  (HX)

Y

F

+ HXY

(1 + Y

F

)

+ (HX)

Y

F

M

+ Y (HX)

M

, where L = log(AHXY + 2), M = max(1, F Y

).

Lemma 4 (see [4]). 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 with 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

.

Lemma 5. Suppose X and Y are large positive numbers, A > 0, α and β are rational numbers (not non-negative integers). Suppose D is a subdomain of {(x, y) | x ∼ X, y ∼ Y } embraced by O(1) algebraic curves, and F = AX

Y

 Y, |a(x)| ≤ 1, |b(y)| ≤ 1. Then

S = X

(x,y)∈D

a

b

e(Ax

y

)

 (XY

+ F

X

Y

+ F

X

Y

+ F

X

Y

+ F

X

Y

+ F

X

Y

) log

F.

P r o o f. This is Theorem 3 of the old version of our paper [9]. The proce- dure of the proof is the same as Theorem 2 of [1]. The difference lies in that we use Lemma 4 above three times to choose parameters optimally and in the last step the exponent pair (1/2, 1/2) is used.

4. Proof of Theorem 2. By our Basic Lemma, we only need to prove that for fixed 1 ≤ M ≤ x

/2, we have

(4.1) S(M ) = X

m∼M

∆

 x m



 x

.

Case 1: M  x

. By the well-known Vorono¨ı formula, we have S(M ) = X

m∼M

x

m

X

n≤x^7/25

d(n) n

cos

 4π √ nx

m − π

4



+ O(x

).

Hence for some 1  N  x

, we have (4.2) x

S(M ) 

X

m∼M

x

m

X

n∼N

d(n) n

e

 2 √ nx m

  

 + x

.

So it suffices to estimate the sum on the right side of (4.2), denoted by S(M, N ).

Let a

= M

m

, b

= d(n)N

n

. Then obviously (4.3) x

S(M, N )  x

M

N

X

m∼M

X

n∼N

a

b

e

 2 √ nx m

   .

We suppose x

 M  x

. For M  x

, we have S(M )  x

by the trivial estimate ∆(t)  t

.

Let T (M, N ) denote the two-dimensional sum on the right side of (4.3).

If N ≥ M , we use Lemma 5 to bound T (M, N ) (take (X, Y ) = (N, M )) and we get

x

T (M, N )  N M

+ x

N

M

(4.4)

+ x

N

M

+ x

N

M

+ x

N

M

+ x

N

M

. Inserting (4.4) into (4.3) we have

x

S(M, N )  (N x)

+ x

M

+ x

N

M

(4.5)

+ x

M

+ x

N

M

+ x

N

M

 x

,

where N ≥ M and M  x

were used.

If x

M

 N < M , we again use Lemma 5 to bound T (M, N ) (whence S(M, N )), but this time we take (X, Y ) = (M, N ), and we get

x

S(M, N )  x

N

+ x

N

M

(4.6)

+ x

N

M

+ x

M

+ x

N

M

+ x

N

M

 x

,

where N < M and M  x

were used.

If N  x

M

, we use the exponent pair (1/6, 4/6) to estimate the sum over m and estimate the sum over n trivially to get

(4.7) x

S(M, N )  x

.

Case 2: x

 M  x

. By the well-known expression

∆(t) = −2 X

n≤t^1/2

ψ

 t n



+ O(1), we have

(4.8) S(M ) = −2 X

m∼M

X

nm≤x^1/2

ψ

 x nm



+ O(x

).

So it suffices to bound the sum S

(M, N ; x) = X

(m,n)∈D

ψ

 x nm

 ,

where D = {(m, n) | m ∼ M , n ∼ N , nm ≤ x

}.

By the well-known finite Fourier expansion of ψ(t) we have S

(M, N ; x)  M N

J + X

h≤J

h

X

(m,n)∈D

e

 hx nm

    (4.9)

 M N

J + X

H

X

h∼H

H

X

(m,n)∈D

e

 hx nm

   , where H runs through {2

| 0 ≤ j ≤ log J/ log 2}. So it suffices to bound

Φ(H, M, N ) = X

h∼H

H

X

(m,n)∈D

e

 hx nm

   .

By Lemma 2 (for details see Liu [4]) we get x

Φ(H, M, N )  N

H

F

X

h∼H

X

(m,r)∈D1

c(m)b(r)e

 2 √ rhx m

   (4.10) 

+ (HF )

+ x

,

where F = x/(N M

), D

is a subdomain of {(m, r) | m ∼ M, r ∼ HF N

}, |c(m)| ≤ 1, |b(r)| ≤ 1.

Now using Lemma 3 to estimate the sum in (4.10) we get (take (h, x, y) = (h, r, m))

x

Φ(H, M, N )  H

F

N

M

+ H

F

M

(4.11)

+ H

F

M

+ H

F

N

M

+ H

F

N

M

+ F

N

M

+ H

F

N

M

+ x

.

Insert (4.11) into (4.9) and then choose a best J ∈ (0, x

). Via Lemma 4 we get

x

S

(M, N ; x)  F

N

M

+ F

N

M

(4.12)

+ F

N

M

+ F

N

M

+ F

N

M

+ F

N

M

+ F

N

M + x

.

Now if we notice that F = x/(N M

), M N  x

and x

 M  x

we obtain

x

S

(M, N ; x)  x

M

+ x

M

+ x

M

(4.13)

+ x

M

+ x

M

+ x

M

+ x

M

+ x

 x

, whence (4.1) is true in this case.

Case 3: x

 M  x

. We use notations of Case 1. Applying Lemma 1 to the sum over m we get

x

S(M, N )  (xN )

M

+ x

 x

if N  M

x

.

Now suppose N  M

x

. Applying Lemma 2 to the variable m (we omit the routine details which can be found in Liu [4]) we get

x

S(M, N )  M N    X

n∼N

X

u∼√ nxM⁻²

c(n)b(u)e(2 √

2 u

(nx)

)   (4.14) 

+ x

,

where c(n)  1, b(u)  1. We apply Lemma 5 to the sum on the right side of (4.14) to get (take (X, Y ) = (N, F/M ))

x

S(M, N )  (xN )

+ x

N

M

(4.15)

+ x

N

M

+ x

N

M

+ x

N

M

+ x

N

M

 x

 x

, where N  x

and M  x

were used.

From our discussions we know (4.1) is true in any case and Theorem 2 follows.

Acknowledgements. The authors thank Prof. Pan Changdong for his

kind encouragement and the referee for his valuable suggestions.

References

[1] R. C. B a k e r and G. H a r m a n, Numbers with a large prime factor, Acta Arith. 73 (1995), 119–145.

[2] E. C o h e n, On the average number of direct factors of a finite abelian group, ibid. 6 (1960), 159–173.

[3] E. K r ¨a t z e l, On the average number of direct factors of a finite Abelian group, ibid.

51 (1988), 369–379.

[4] H. Q. L i u, The distribution of 4-full numbers, ibid. 67 (1994), 165–176.

[5] —, On the number of abelian groups of a given order (supplement), ibid. 64 (1993), 285–296.

[6] —, On some divisor problems, ibid. 68 (1994), 193–200.

[7] H. M e n z e r, Exponentialsummen und verallgemeinerte Teilerprobleme, Habilita- tionsschrift, FSU, Jena, 1992.

[8] P. G. S c h m i d t, Zur Anzahl unit¨arer Faktoren abelscher Gruppen, Acta Arith. 64 (1993), 237–248.

[9] W. G. Z h a i and X. D. C a o, On the average number of direct factors of finite abelian groups, ibid. 82 (1997), 45–55.

Department of Mathematics Beijing Institute

Shandong Normal University of Petrochemical Technology

Jinan, Shandong 250014 Daxing, Beijing 102600

P.R. China P.R. China

E-mail: arith@sdunetnms.sdu.edu.cn E-mail: biptiao@info.iuol.cn.net Received on 29.7.1996

and in revised form on 15.7.1997 (3030)