Divisor problems of 4 and 3 dimensions by

LXXIII.3 (1995)

Divisor problems of 4 and 3 dimensions

by

Hong-Quan Liu (Harbin)

Introduction. For a positive integer n, let the divisor functions d(4, 5, 6, 7; n), d(1, 1, 2, 2; n) and d(1, 1, 2; n) be defined as in [3], [4]. In this paper we will sharpen our former arguments by proving the following new results regarding the errors of distribution of these divisor functions. We have (ε and x are as usual):

Theorem 1.

X

n≤x

d(4, 5, 6, 7; n) = main terms + O(x

).

Theorem 2.

X

n≤x

d(1, 1, 2, 2; n) = main terms + O(x

).

Theorem 3.

X

n≤x

d(1, 1, 2; n) = main terms + O(x

).

Let Q

(x) be the number of 4-full numbers not exceeding x, let τ (G) be the number of direct factors of a finite Abelian group G, and t(G) be the number of unitary factors of G, and T (x) = P

τ (G), T

(x) = P t(G), where the summations are over all G of order not exceeding x. Then, as in [3], [4], we have

Corollary 1. Q

(x) = main terms + O(x

).

Corollary 2. T (x) = main terms + O(x

).

Corollary 3. T

(x) = main terms + O(x

).

Note that 87/869 = 0.1001150 . . . , which improves the corresponding exponent 6/59 = 0.10169 . . . established in Theorem 2 of [3], and 7/19 = 0.3684 . . . , 29/80 = 0.3625 improve respectively the exponents 0.4 and 77/208 = 0.3701 . . . given by Theorems 2 and 1 of [4].

[249]

In demonstrating these theorems, Theorem 3 of [1] will again play an important role. We will also need to combine other tools existing in papers [2] to [5] of the author. Needless to say, many tedious and elementary cal- culations will emerge in our treatment, which is inherent in such divisor problems. We will do our best to avoid redundancy.

1. Proof of Theorem 1. We recall a useful lemma (Theorem 3 of [1]).

Lemma 1.1. Let H ≥ 1, X ≥ 1, Y ≥ 1000; let α, β and γ be real numbers with αγ(γ − 1)(β − 1) 6= 0, and let A > C(α, β, γ) > 0 and 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 that |C

(h, x)| ≤ 1, |C

(y)|

≤ 1 and F = AH

X

Y

 Y . Then, for L = ln((A + 1)HXY + 2) and M = max(1, F Y

),

L

S(H, X, Y ) 

√

(HX)

Y

F

+ HXY

(1 + Y

F

)

+

√

(HX)

Y

F

M

+ √

(HX)

Y

M .

We stress that the condition F  Y is needed in the proof of this lemma.

We adopt the notations introduced in [3]. In particular, from (7) of [3], we have the following estimate:

Φ(H; N )  H

(N

H

G

)

X

h∼H

   X

5

g

(n

)g

(n

)g

(u)e(g)   (1.1) 

+ N

(HG)

ln x + x

.

From (23) to (31) of [3], and the estimates on p. 175 there (η = ε/8), x

S(a, b, c, d; N ) 

√

x

N

N

+

√

x

N

N

(1.2)

+

√

x

N

N

+

√

x

N

N

+ x

. We need two more estimates for S(a, b, c, d; N ). First we employ Lemma 1.1 to the triple summation over n

, n

and u in (1.1), with the choice (h, x, y)

= (n

, n

, u). Note that U ∼ = HG/N

; this yields x

Φ(H; N ) 

√

(HG)

N

N

N

+ √

HG(N

N

)

N

+

√

(HG)

(N

N

)

N

+

√

(HG)

(N

N

)

N

+

√

(HG)

(N

N

)

N

+ √

(HG)

N

N

+ √

HG(N

N

)

N

+ x

.

We put the above estimate in (1) of [3] and choose the parameter K opti- mally via a well-known lemma (cf. Lemma 3 of [3]) to get

x

S(a, b, c, d; N ) 

√

G

(N

N

)

N

+ √

G(N

N

)

N

(1.3)

+

√

G

(N

N

)

N

+

√

G

(N

N

)

N

+

√

G

(N

N

)

N

+ √

GN

N

N

+ √

G

N

N

N

+ x

√

x

N

N

N

+

√

xN

N

N

(1.4)

+

√

x

N

N

N

+

√

x

N

N

N

+

√

xN

N

N

+

√

xN

N

N

+ x

. To pass from (1.3) to (1.4) we have invoked (18) of [3]. By (21) of [3], (1.5) x

S(a, b, c, d; N )  √

xN

N

N

+ x

. From (1.4) and (1.5) we infer that

(1.6) x

S(a, b, c, d; N )  X

1≤i≤6

P

+ x

,

where

P

= min(

√

x

N

N

N

, √

xN

N

N

) 

√

x

N

N

, (1.7)

P

= min(

√

xN

N

N

, √

xN

N

N

) 

√

x

N

N

, (1.8)

P

= min(

√

x

N

N

N

, √

xN

N

N

) ≤

√

x

N

N

, (1.9)

P

= min(

√

x

N

N

N

, √

xN

N

N

) (1.10)

≤

√

x

N

N

, P

= min(

√

xN

N

, √

xN

N

)  x

, (1.11)

P

= min(

√

xN

N

N

, √

xN

N

N

) 

√

x

N

N

. (1.12)

Next, we again apply Lemma 1.1 to the triple summation over n

, n

and u in (1.1), but with the choice (h, x, y) = (n

, u, n

). This gives

x

Φ(H; N ) 

√

(HG)

N

N

N

+ (HG)

N

N

+ (HG)

N

N

+

√

(HG)

N

N

N

+

√

(HG)

N

N

N

+ √

HGN

N

N

+ √

(HG)

N

N

N

+ x

.

We put the above estimate in (1) of [3] and choose K optimally to get

x

S(a, b, c, d; N ) 

√

G

N

N

N

+

√

G

N

N

N

(1.13)

+

√

G

N

N

N

+

√

G

N

N

N

+

√

G

N

N

N

+ √

GN

N

N

+ √

G

N

N

N

+ x

√

x

N

N

N

+

√

xN

N

+

√

xN

N

+

√

x

N

N

N

+

√

x

N

N

+

√

xN

N

N

+

√

xN

N

N

+ x

. From (1.5) and (1.13) we get

x

S(a, b, c, d; N )  x

+ X

1≤i≤4

Q

+

√

xN

N

(1.14)

+

√

xN

N

+

√

x

N

N

, where

Q

= min(

√

x

N

N

N

, √

xN

N

N

) (1.15)

≤

√

x

N

N

, Q

= min(

√

x

N

N

N

, √

xN

N

N

) (1.16)

≤

√

x

N

N

, Q

= min(

√

xN

N

N

, √

xN

N

N

) ≤

√ xN

, (1.17)

Q

= min(

√

xN

N

N

, √

xN

N

N

) ≤ x

. (1.18)

From (31) of [3] we have

(1.19) x

S(a, b, c, d; N ) 

√

(x(N

N

)

)

+ x

. By (1.2) and (1.19) we have

(1.20) x

S(a, b, c, d; N )  X

1≤i≤4

R

+ x

,

where

R

= min(

√

x

N

N

,

√

(x(N

N

)

)

) ≤

√ x

N

, (1.21)

R

= min(

√

x

N

N

,

√

(x(N

N

)

)

) ≤

√

x

N

, (1.22)

R

= min(

√

x

N

N

,

√

(x(N

N

)

)

) ≤

√

x

N

, (1.23)

R

= min(

√

x

N

N

,

√

(x(N

N

)

)

) ≤

√

x

N

.

(1.24)

From (1.20) to (1.24), we find that the required estimate follows if N

≤ x

. We assume hereafter that N

> x

. From (1.19) and (1.6) to (1.12) we have

(1.25) x

S(a, b, c, d; N ) 

√

x

N

N

+ X

1≤i≤4

S

+ x

, where

S

= min(

√

x

N

N

,

√

(x(N

N

)

)

) (1.26)

≤

√

x

N

< x

, S

= min(

√

x

N

N

,

√

(x(N

N

)

)

) (1.27)

≤

√

x

N

≤ x

, S

= min(

√

x

N

N

,

√

(x(N

N

)

)

) (1.28)

≤

√

x

N

≤ x

, S

= min(

√

x

N

N

,

√

(x(N

N

)

)

) (1.29)

≤

√

x

N

≤ x

. From (1.25) to (1.29) we have

(1.30) x

S(a, b, c, d; N ) 

√

x

N

N

+ x

. By (1.30) and (1.14) to (1.18) we have

(1.31) x

S(a, b, c, d; N )  X

1≤i≤6

T

+ x

, where

T

= min(

√

x

N

N

,

√

x

N

N

) (1.32)

≤ (x

N

)

≤ x

, T

= min(

√

x

N

N

,

√

x

N

N

) (1.33)

≤ (x

N

)

≤ x

, T

= min(

√

x

N

N

,

√

xN

) ≤ (x

N

)

≤ x

, (1.34)

T

= min(

√

x

N

N

,

√

xN

N

) ≤ (x

N

)

≤ x

, (1.35)

T

= min(

√

x

N

N

,

√

xN

N

) (1.36)

≤ (x

N

)

≤ x

, T

= min(

√

x

N

N

,

√

x

N

N

) (1.37)

≤ (x

N

)

≤ x

.

By (1.30) to (1.37), we have completed the proof.

2. Proof of Theorem 2. Let (a, b, c, d) be any permutation of (1, 1, 2, 2).

It suffices to obtain

(2.1) S(a, b, c, d; N )  x

,

where η = ε/8, N = (N

, N

, N

), N

, N

and N

are positive integers with (2.2) N

 N

 N

, N

N

N

 x, N

N

N

> x

,

and the sum S(a, b, c, d; N ) is defined on p. 199 of [4]. We will retain many familiar notations used in both [3] and [4].

The case of (a, b, c, d) = (1, 1, 2, 2) can be dealt with immediately. In fact, from (2.2) we have N

N

 x

, thus (GN

N

N

)

 (xN

N

)

 x

, and the required estimate follows from Lemma 6 of [4].

For (a, b, c, d) = (1, 2, 1, 2), by (2.2) we have N

N

 N

N

N

 x, thus again (GN

N

N

)

 x

, and the required estimate follows.

We now show

(2.3) S(2, 1, 1, 2; N )  x

.

To this end, we first proceed similarly to pp. 167–170 of [3]. This yields, similarly to (7) of [3],

Φ(H; N )  H

(N

H

G

)

X

h∼H

   X

1

F (n

)R(n

)S(u)e(g)   (2.4) 

+ N

(HG)

ln x + x

(for an explanation of the error term x

, cf. p. 199 of [4]), where P means summation over n

, n

and u with

1 < n

< n

, N

≤ n

< 2N

(v = 1, 2), U

< u < U

,

and G, U

, U

and the function g are defined on p. 169 of [3]. In particular, g = C

(xn

n

h

u)

. Moreover, F (·), R(·), S(·) are suitable monomials with absolute values ∼ = 1. We can apply Lemma 1 of [3] one more time, to the variable n

of (2.4). We have

Φ(H; N )  N

N

(H

G)

(2.5)

× X

h∼H

X

n1∼N1

 X

U1<u<U2

X

V1<v<V2

T (u)Q(v)e(g

)    + N

(HG)

ln x + x

,

where V

= V

(h, n

, u) (i = 1, 2), |T (u)| ≤ 1, |Q(v)| ≤ 1, and g

= C

(h

xuvn

)

. We can relax the condition U

< u < U

to u ∼ = U :=

HGN

and the condition V

< v < V

to v ∼ = V := HGN

consecutively

by means of Lemma 5 of [3] (note that we can assume that x is quadratic

irrational, cf. p. 168 of [3]); we thus deduce from (2.5) that

x

Φ(H; N )  N

N

(H

G)

(2.6)

× X

h∼H

X

n1∼N1

   X

u∼=U

X

v∼=V

T

(u)Q

(v)e(g

)    + N

(HG)

+ x

,

where |T

(u)|, |Q

(v)| ≤ 1. From (2.6) it is evident that x

Φ(H; N )  N

N

(H

G)

(2.7)

× X

h∼H

X

n₁∼N₁

   X

w∼=W

K(w)e(C

(h

xwn

)

)    + N

(HG)

+ x

,

where W = U V = (HG)

N

N

and |K(w)| ≤ 1.

If HG  N

N

, we apply Lemma 1.1 to the triple exponential sum in (2.7), with (h, x, y) = (h, n

, w), to get

x

Φ(H; N ) 

√

H

G

N

(N

N

)

+ (HG)

N

(N

N

)

(2.8)

+ (HG)

N

(N

N

)

+

√

H

G

N

(N

N

)

+

√

H

G

N

(N

N

)

+ √

H

G

N

+ √

GN

(N

N

)

+ x

√

G

N

(N

N

)

+ N

(N

N

)

+

√

G

N

(N

N

)

+ √

GN

(N

N

)

+ N

(HG)

+

√

H

G

N

(N

N

)

+ √

H

G

N

+ x

.

If N

≥ x

, we have (GN

N

N

)

= (xN

N

)

 (x

N

)

 x

, and (2.3) follows from Lemma 6 of [4]. We now assume that N

<

x

. Then we easily see that the total contribution of the first four terms in (2.8) is  x

. In fact, since N

N

N

 x

,

22

√

G

N

(N

N

)

√

x

N

(N

N

)

√

x

N

 x

, N

(N

N

)

√

x

N

 x

,

32

√

G

N

(N

N

)

√

x

N

(N

N

)

√

x

N

 x

,

√

GN

(N

N

)

 √

xN

(N

N

)

√

x

N

 x

. Thus from (2.8) we get

x

Φ(H; N )  N

(HG)

+

√

H

G

N

(N

N

)

(2.9)

+ √

H

G

N

+ x

.

If HG  N

N

, we go back to the original definition for Φ(H; N ), and we produce a new integral variable q from n

and n

such that q = n

n

. Since HG  N

N

, Lemma 1.1 is applicable with (h, x, y) = (h, n

, q), and we get

x

Φ(H; N ) 

√

G

N

(N

N

)

+ N

(N

N

)

+ N

(HG)

+

√

H

G

N

(N

N

)

+

√

G

N

(N

N

)

+ √

H

G

N

+ √

GN

(N

N

)

+ x

 N

(HG)

+

√

H

G

N

(N

N

)

+ √

H

G

N

+ x

.

Thus we see that (2.9) always holds. We put the estimate (2.9) in (1) of [3]

and choose the parameter K optimally via Lemma 3 of [3] to get x

S(2, 1, 1, 2; N ) 

√

G

N

(N

N

)

+

√

G

N

(N

N

)

+ √

G

N

(N

N

)

+ x

√

x

N

(N

N

)

+

√

x

N

(N

N

)

+ √

x

N

N

N

+ x

√

x

N

+

√

x

N

+

√

x

N

+ x

 x

, which proves (2.3).

We proceed to estimate S(2, 2, 1, 1; N ); the remaining two cases with (a, b, c, d) = (2, 1, 2, 1) and (1, 2, 2, 1) can be treated similarly. As in (7) of [3], we get

Φ(H; N )  H

(N

H

G

)

(2.10)

× X

h∼H

X

2

g

(n

)g

(n

)g

(u)e

 C

 xhu n

n



   

+ N

N

ln x + (Hx(N

N

N

)

)

+ x

, where P

2

means summation over lattice points (n

, n

, u) such that 1 ≤ n

< n

, n

∼ N

, n

∼ N

,

hx(n

n

M

)

< u < hx(n

n

M

)

,

and where M

= max(N

, n

), M

= min((xn

n

)

, 2N

), g

(·) are

monomials with |g

(·)| ∼ = 1. By an appeal to Lemma 5 of [3], we can relax

the summation range to u ∼ = U = HGN

. Then we can produce a double

sum in (2.10) by setting hu = r and n

n

= s. This yields