(x) denote the number of cube-full numbers in the interval [1, x], where x is a sufficiently large number. On the one hand, an asymptotic formula for Q

LXVII.1 (1994)

The number of cube-full numbers in an interval

by

Hong-Quan Liu (Harbin)

1. Introduction. A positive integer n is called cube-full if p | n implies that also p

| n; here p denotes a prime number. Let Q

(x) denote the number of cube-full numbers in the interval [1, x], where x is a sufficiently large number. On the one hand, an asymptotic formula for Q

(x) can be given with an error term o(x

), and this order of magnitude cannot be reduced to O(x

) with some µ < 1/8 without any hypothesis. On the other hand, an asymptotic formula does hold for Q

(x + x

) − Q

(x) with a µ < 1/8 by using results for exponential sums. In fact, Shiu [7] proved that

(∗) Q

(x + x

) − Q

(x) → Cx

, for any 140

1123 < µ < 1 3 ;

here 140/1123 = 0.1246607 . . . (< 1/8). The aim of this paper is to give an improvement on his result. We will prove

Theorem 1. (∗) holds for 1

3 > µ > 11

92 = 0.1195652 . . .

Taking the idea from the author’s paper [3] dealing with squarefull in- tegers, we first give a reduction of our problem, which connects (∗) with some exponential sums, but this time (in contrast to [3]) triple exponen- tial sums are inevitable, and, to obtain our result, it is crucial to apply the method developed in [2] and [4]. Actually, we also give an improvement of Kr¨atzel’s result about ∆(x; 3, 4, 5), where ∆(x; 3, 4, 5) is the error term in the asymptotic formula

X

x≥r³s⁴t⁵

1 = A

x

+ A

x

+ A

x

+ ∆(x; 3, 4, 5),

A

= Y

3≤n≤5 n6=i

ζ

 n i



, i = 3, 4, 5.

[1]

Our estimates then imply ∆(x; 3, 4, 5)  x

, where ε is an arbitrar- ily small positive number, while [1] finds ∆(x; 3, 4, 5)  x

, with 22/177 = 0.1242937 . . .

2. Reduction. We prove Theorem 2. Let

S

(x) = X

(m,n)∈D

ψ

 x m

n





, ψ(ξ) = ξ − [ξ] − 1 2 ,

where (a, b, c) is any permutation of (3, 4, 5), D is the range {(m, n) | m

n

≤ x, m ≥ n} or {(m, n) | m

n

≤ x, m ≥ x

} or {(m, n) | n ≤ m ≤ x

}. Then

S

(x)  x

implies the assertion of Theorem 1.

P r o o f. Put B = x

, and suppose that 11/92 + ε < µ < 1/3. It is well known that

Q

(x) = X

a³b⁴c⁵∈I b,c≤B

|µ(bc)| + X

a³b⁴c⁵∈I b>B or c>B

|µ(bc)| = X

1

+ X

2

, say, where I is the interval (x, x + x

]. We have

X

1

= ((x + x

)

− x

) X

b≤B

|µ(b)|

b

X

(c,b)=1 c≤B

|µ(c)|

c

+ O(B

), X

(c,b)=1 c≤B

|µ(c)|

c

= X

|µ(c)|

c

+ O(B

)

= ζ(5/3) ζ(10/3)

Y

p|b

 1 + 1

p



+ O(B

), X

b≤B

|µ(b)|

b

Y

p|b

 1 + 1

p



= Y

p

 1 + 1

p

 1 + 1

p





+ O(B

),

(x + x

)

− x

= 1

3 x

(1 + O(x

)), and thus

X

1

= Cx

(1 + o(1)), C = 1

3 · ζ(5/3) ζ(10/3)

Y

p

 1 + 1

p

 1 + 1

p





.

It remains to estimate P

2

. First we consider the portion of P

2

with b > B, which we denote by P

21

. We have, with x

= x + x

and a(m) = P

m=a³c⁵

1, X

21

≤ X

B<b≤x^1/12₁

X

(xb⁻⁴)^1/3<ac^5/3≤(x₁b⁻⁴)^1/3

1 (∗∗)

+ X

m≤x^2/3₁

a(m) X

(xm⁻¹)^1/4<b≤(x₁m⁻¹)^1/4

1.

At this stage, to treat the sums of (∗∗) we need to cite the next two lemmata.

Lemma 2.1. For y ≥ 1, % > 0, 1 ≤ µ ≤ y

≤ v, µ

v = y, X

y^1/(%+1)<n≤v

ψ



%

r y n



= X

µ<n≤y^1/(%+1)

ψ

 y n



− %ψ

(µ)yµ

+ O(yµ

+ 1), where ψ

(z) :=

ψ(z)

− 1/24.

Lemma 2.2. With the same assumptions as in Lemma 2.1, X

m^%n≤y

1 = ζ(%)y +ζ

 1

%



y

− X

n≤µ

ψ

 y n



− X

n≤v

ψ



%

r y n



−%ψ

(µ)yµ

+ O(yµ

) + 1/4 − ψ(µ)ψ(v) + O(µv

).

Lemma 2.1 is Hilfssatz 4 of P. G. Schmidt [6], and Lemma 2.2 is (8) on p. 37 of [6]. Now by Lemma 2.2 we have, with µ = (x

b

)

,

X

ac^5/3≤(x₁b⁻⁴)^1/3

1 = ζ

 5 3



(x

b

)

+ ζ

 3 5



(x

b

)

− X

n≤µ

ψ((x

b

n

)

)

− X

n≤µ

ψ((x

b

n

)

) + O(1)

and a similar formula holds for x

being replaced by x. Thus the first sum of (∗∗) is (note that we can assume, for example, µ < 0.125, in light of [7])

≤ O(B

x

) + S

,

where S

is a linear combination of S

(Ω), Ω = x or x

, and (a, b, c) be-

longs to the set {(3, 4, 5), (3, 5, 4), (5, 4, 3), (5, 3, 4)}. To deal with the second

sum in (∗∗) we apply the technique in Schmidt [6]. First it is easy to observe

that X

m≤x^2/3₁

a(m) X

(xm⁻¹)^1/4<b≤(x₁m⁻¹)^1/4

1 ≤ I

+ I

, where

I

= X

a³c⁵≤x^2/3₁ ,a≤x^1/12₁

X

(xa⁻³c⁻⁵)^1/4<b≤(x₁a⁻³c⁻⁵)^1/4

1,

I

= X

a³c⁵≤x^2/3₁ ,c≤x^1/12₁

X

(xa⁻³c⁻⁵)^1/4<b≤(x₁a⁻³c⁻⁵)^1/4

1.

It is easy to see that

I

= O(x

) + S(x

) − S(x), S(x

) = X

a≤x^1/12₁

X

c≤(x^2/3₁ a⁻³)^1/5

ψ

 x

a

c





and S(x) is defined similarly. We have X

c≤(x^2/3₁ a⁻³)^1/5

ψ

 x

a

c





=

 X

c≤(x1a⁻³)^1/9

+ X

(x1a⁻³)^1/9<c≤(x^2/3₁ a⁻³)^1/5

 ψ(·)

= I

+ I

, say.

From I

we can get sums of the type S

(x

) easily, and for I

we choose y = (x

a

)

, % = 4/5, µ = x

, v = (x

a

)

in Lemma 2.1, to obtain

I

= X

x^1/12₁ <n≤(x1a⁻³)^1/9

ψ

 x

a

n





+ O(yµ

+ 1).

Thus from I

we also get sums of the type S

(Ω), with a permissible error. Similarly we can treat I

. The other portion of P

2

with the condition c > B can be treated along the same lines by using Lemmata 2.1 and 2.2.

Hence our problem is reduced to treating a linear combination of sums of the type S

(Ω). The proof of Theorem 2 can thus been finished.

3. Three general estimates for S

(x). Clearly we can assume that D = {(m, n) | m

n

≤ x, m ≥ n};

the other two cases can be treated similarly and more easily. We need

Lemma 3.1. 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 for A > C(α, β, γ) > 0, let 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 } (h ∼ H means that H ≤ h < 2H, etc.) 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 ) 

p

(HX)

Y

F

+ HXY

(1 + Y

F

)

+

p

(HX)

Y

F

M

+ p

(HX)

Y

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

).

Lemma 3.2. Let f (x, y) be an algebraic function in the rectangle D

= {(x, y) | x ∼ X, y ∼ Y }, f (x, y) = Ax

y

for (x, y) ∈ D

, D be a subdomain of D

bounded by O(1) algebraic curves. Suppose that X  Y , N = XY , A > 0, F = AX

Y

, and αβ(α + β − 1)(α + β − 2) 6= 0. Then

S := (N F )

X

(x,y)∈D

e(f (x, y))

 √

F

N

+ N

+ √

N

F

X

+ N F

+ N Y

.

Lemma 3.1 is Theorem 3 of [4], and Lemma 3.2 is Lemma 9 of [2]. We also need

Lemma 3.3. Let f (x) and g(x) be algebraic functions in the interval [a, b], and

|f

(x)| ∼ = R

, |f

(x)|  (RU )

,

|g(x)| ≤ H, |g

(x)|  HU

, U, U

≥ 1.

Then X

a≤n≤b

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

α≤u≤β

b

g(n(u))

p f

(n(u)) e(f (n(u)) − un(u) + 1/8) + O(H ln(β − α + 2) + H(b − a + R)(U

+ U

)) + O(H min(R

, max(1/hαi, 1/hβi))),

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

(x), n(u) is

determined by the equation f

(n(u)) = u, b

= 1/2 or 1 according as u is

one of α, β or not, hxi is defined by hxi =

 kxk if x is not an integer , kxk = min

|x − n|, n ∈ Z, β − α if x is an integer ,

and √

f

> 0 if f

> 0, √

f

= i p

|f

| if f

< 0.

P r o o f. This is Theorem 2.2 of [5].

Now we proceed to deal with S

(x). It suffices to estimate S(M, N ), where

S(M, N ) = X

(m,n)∈D

ψ((xm

n

)

),

D = D(M, N ) = {(m, n) | m ∼ M, n ∼ N, m

n

≤ x, m ≥ n}, and M , N are any positive integers such that

M N > x

, 2M ≥ N, M

N

≤ x.

Then, using the familiar reduction (cf. [2]), for a parameter K ∈ [100, M N ], we get, with η = ε

, and some H ≤ K

,

(0) x

S(M, N )  M N K

+ min(1, K/H)Φ(H, M, N ), where

Φ(H, M, N ) = H

X

h∼H

 X

(m,n)∈D

e(f (h, m, n))   ,

f (h, m, n) = h

 x m

n



.

At this stage, we can assume that x is irrational. We apply Lemma 3.3 to the summation over m, and get, with M

= max(M, n), M

= min((xn

)

, 2M ),

X

M1≤m≤M2

e(f (h, m, n))

= X

U1<u<U2

C

(x

h

n

u

)

e(C

(xh

u

n

)

)

+ O

 M

HF + ln x

 + O

 min

 M

HF



, 1

U

− hb/a



+ X

1≤i≤2

min

 M

HF



, 1

kg(n, X

)k



+ R(h, n), where C

, C

, C

, . . . denote certain constants,

U

= hb

a (xn

M

)

, U

= hb

a (xn

M

)

,

g(n, X) = hb

a (xn

X

)

,

X

= max(n, M ), X

= 2M, F = (xM

N

)

, R(h, n) =

 



1

2

C

(x

h

n

u

)

e(C

(xh

u

n

)

) for M

= (xn

)

, u = U

= hb/a;

0 otherwise.

It is easy to see that X

n∼N

min

 M

HF



, 1

U

− hb/a



 x

ln x, X

n∈I

R(n, h) 

 M

HF





N

(F HN

)

+ N F H



 x

(note that F  M, . . .), where I is some subinterval of [N, 2N ]. We consider

Ω = X

n∼N

min

 M

HF



, 1

kg(n, X

)k



=

 X

n≤M,n∼N

+ X

n>M,n∼N



min(A, B) = Ω

+ Ω

, where

Ω

= X

n≤M,n∼N

min

 M

HF



, 1

kg

(n)k

 ,

g

(n) = hb

a (xn

M

)

, Ω

= X

n>M,n∼N

min

 M

HF



, 1

kg

(n)k

 ,

g

(n) = hb

a (xn

)

= hb

a (xn

)

.

As g

(y) is monotonic, and g

(y) ∼ = H(xN

M

)

for y ∼ N , by Hilfssatz 4 of Kr¨atzel [1] we have

Ω

 (1 + H(xN

M

)

)

× ((M

H

F

)

+ H

(x

N

M

)

ln x)

 (HF )

+ x

ln x

and similarly, Ω

 (HF )

+x

ln x (note that Ω

6= 0 implies N  M ),

and

X

n∼N

min

 M

HF



, 1

kg(n, X

)k



 (HF )

+ x

ln x.

From the above observations, we achieve, after a double Abelian sum- mation, the estimate

Φ(H, M, N )  H

 M

HF



X

h∼H

 X

(u,n)∈D⁰

e(C

(xh

u

n

)

)    (1)

+ (HF )

+ x

ln x,

where D

is a range contained in {(u, n) | n ∼ N, C

≤ HF/(uM ) ≤ C

}, bounded by O(1) algebraic curves. We will apply the next lemma to choose parameters optimally.

Lemma 3.4. 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 nonnegative numbers, Q

≤ Q

. Then there exists an Q ∈ [Q

, Q

] with

X

1≤m≤M

A

Q

+ X

1≤n≤N

B

Q

 X

1≤m≤M

X

1≤n≤N

(A

B

)

+ X

1≤m≤M

A

Q

+ X

1≤n≤N

B

Q

. This is Lemma 2 of [2]. Now if we apply Lemma 3.2 to the inner double sum in (1), put our estimate into (0), and choose K optimally via Lemma 3.4, we get

Lemma 1.

x

S(M, N ) 

√

x

M

N

+

√

x

M

N

+ x

. If we choose (h, x, y) = (h, u, n) in Lemma 3.1, then we get an estimate for the triple exponential sum in (1). Putting it into (0), and choosing K optimally via Lemma 3.4, we immediately get

Lemma 2.

x

S(M, N ) 

√

F

M

N

+

√

F

M

N

+

√

F

M

N

+

√

F

M

N

+

√

F

M

N

+ √

F

M

N

+ √

F M N

+ x

.

To obtain our last general estimate, we apply Lemma 3.3 to the summa- tion over n in (1), to get

Φ(H, M, N )  M N H

F

X

h∼H

 X

(u,v)∈D⁰⁰

P (u)Q(v)e(G(u, v, h))   (2) 

+ (HF )

+ x

ln x,

where |P (u)| ≤ 1, |Q(v)| ≤ 1, and D

is a suitable domain contained in the rectangle {(u, v) | HF/(M u) ∈ [C

, C

], HF/(N v) ∈ [C

, C

]}. We then apply Lemma 3.1 to the sum of (2) by choosing (h, x, y) = (h, u, v), and taking K optimally via Lemma 3.4, we get

Lemma 3.

x

S(M, N ) 

√

F

M

N

+

√

F

M

N

+

√

F

M

N

+

√

F

M

N

+

√

F

M

N

+ √

F

M

N

+ √

F

M

N

+ x

.

4. Proof of Theorem 1. As M  N , we easily observe that F  (xM

N

)

,

thus by Lemmata 1 and 3 we obtain (3) x

S(M, N ) 

√

x

M

N

+

√

x

N

+ x

, (4) x

S(M, N ) 

√

x

M

N

+

√

x

M

N

+

√

x

M

N

+

√

x

M

N

+

√

x

M

N

+

√

x

M

N

+

√

x

M

N

+ x

, and thus

x

S(M, N )  X

1≤i≤11

R

+

√

x

M

N

+

√

x

M

N

+ √

xM

N

+ x

, where

R

= min(

√

x

M

N

,

√

x

M

N

)

 (

√

x

(M

N

)

)

(

√

x

M

N

)

= x

, R

= min(

√

x

M

N

,

√

x

M

N

)

 (

√

xM

N )

(

√

x

M

N

)

= x

< x

, R

= min(

√

x

M

N

,

√

x

M

N

)

 (

√

x(M

N )

)

(

√

x

M

N

)

= x

< x

, R

= min(

√

x

M

N

,

√

x

M

N

)

 (

p

x(M

N )

)

(

√

x

M

N

)

= x

< x

,

R

= min(

√

x

N

,

√

x

M

N

)

 (

√

xN

)

(

√

x

N

)

= x

< x

, R

= min(

√

xN

,

√

x

M

N

)

 (

√

xN

)

(

√

x

N

)

= x

< x

, R

= min(

√

xN

,

√

x

M

N

)

 (

√

xN

)

(

√

x

N

)

= x

< x

, R

= min(

√

xN

,

√

x

M

N

)

 (

√

xN

)

(

√

x

N

)

= x

< x

, R

= min(

√

xN

,

√

x

M

N

)

 (

√

xN

)

(

√

x

N

)

= x

< x

, R

= min(

√

xN

, √

xM

N

)

 (

√

xN

)

( √

xN

)

= x

, R

= min(

√

xN

,

√

x

M

N

)

 (

√

xN

)

(

√

x

N

)

= x

. Hence we have, with θ = 11/92,

x

S(M, N ) 

√

x

M

N

+

√

x

M

N

(5)

+ √

xM

N

+ x

, which, in conjunction with (3), gives

(6) x

S(M, N ) 

√

x

M

N

+ R

+ R

+ R

+ x

√

x

M

N

+ x

. By Lemma 2 we have

x

S(M, N ) 

√

x

M

N

+

√

x

M

N

+

√

x

M

N

(7)

+

√

x

M

N

+

√

xM

N

+ x

. Note that actually

√

x

M

N

 √

xM

N

, thus from (5) we get (8) x

S(M, N ) 

√

x

M

N

+ √

xM

N

+ x

.

From (7) and (8) we get x

S(M, N ) 

√

x

M

N

+

√

x

M

N

+ X

1≤i≤6

S

+ x

, where

S

= min(

√

x

N

,

√

x

N

)

≤ (

√

x

N

)

(

√

x

N

)

= x

< x

, S

= min(

√

x

N

, √

xN

)

≤ (

√

x

N

)

( √

xN

)

= x

< x

, S

= min(

√

x

N

,

√

x

N

)

≤ (

√

x

N

)

(

√

x

N

)

= x

< x

, S

= min(

√

x

N

, √

xN

)

≤ (

√

x

N

)

( √

xN

)

= x

< x

, S

= min(

√

xN

,

√

x

N

)

≤ (

√

xN

)

(

√

x

N

)

= x

< x

, S

= min(

√

xN

, √

xN

) ≤ (

√

xN

)

( √

xN

)

= x

< x

. Thus we have

(9) x

S(M, N ) 

√

x

M

N

+

√

x

M

N

+ x

. From (6), (8) and (9) we finally achieve

x

S(M, N )  X

1≤i≤4

K

+ x

, where

K

= min(

√

x

M

N

,

√

x

M

N

,

√

x

M

N

)

= min(A

, B

, C

) ≤ A

B

C

= x

, K

= min(

√

x

M

N

,

√

x

M

N

,

√

x

M

N

)

= min(A

, B

, C

) ≤ A

B

C

= x

,

K

= min(

√

x

M

N

, √

xM

N

,

√

x

M

N

)

= min(A

, B

, C

) ≤ A

B

C

= x

, K

= min(

√

x

M

N

, √

xM

N

,

√

x

M

N

)

= min(A

, B

, C

) ≤ A

B

C

= x

. This completes the proof.

References

[1] E. K r ¨a t z e l, Zweifache Exponentialsummen und dreidimensionale Gitterpunktpro- bleme, in: Elementary and Analytic Theory of Numbers, Banach Center Publ. 17, PWN–Polish Scientific Publishers, Warszawa, 1985, 337–369.

[2] H.-Q. L i u, On the number of abelian groups of a given order, Acta Arith. 59 (1991), 261–277.

[3] —, The number of squarefull numbers in an interval, ibid. 64 (1993), 129–149.

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

[5] S. H. M i n, Methods in Number Theory, Vol. 2, Science Press, 1981 (in Chinese).

[6] P. G. S c h m i d t, Zur Anzahl Abelscher Gruppen gegebener Ordnung, J. Reine Angew.

Math. 229 (1968), 34–42.

[7] P. S h i u, The distribution of cube-full numbers, Glasgow Math. J. 33 (1991), 287–295.

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Received on 26.3.1993 (2406)