On the number of coprime integer pairs within a circle

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XC.1 (1999)

On the number of coprime integer pairs within a circle

by

Wenguang Zhai (Jinan) and Xiaodong Cao (Beijing)

1. Introduction. Let P (x) denote the number of integer pairs within the circle a

²

+ b

²

≤ x, and E(x) denote the difference P (x) − πx. Then the well-known circle problem is to estimate the upper bound of E(x) and the best result at present is

(1.1) E(x) = O(x

^23/73+ε

).

See Huxley [6].

Let V (x) denote the number of coprime integer pairs within the circle a

²

+ b

²

≤ x. It is an exercise to deduce that

(1.2) V (x) = 6

π x + O(x

^1/2

exp(−c log

^3/5

x(log log x)

^−2/5

)),

where c is some absolute constant. The problem of reducing the exponent 1/2 is open. One way to make progress is to assume the Riemann Hypothesis (RH). W. G. Nowak [11] proved that RH implies

(1.3) V (x) = 6

π x + O(x

^15/38+ε

).

D. Hensley [5] also got a result of this type, but with a larger exponent.

The aim of this paper is to further improve this result. We have the following

Theorem. If RH is true, then

(1.4) V (x) = 6

π x + O(x

^11/30+ε

).

Notations. e(x) = exp(2πix). m ∼ M means c

₁

M ≤ m ≤ c

₂

M for absolute constants c

1

and c

2

. E(x) always denotes the error term in the circle problem. ε denotes an arbitrary small positive number and may be different at each occurrence.

1991 Mathematics Subject Classification: 11N37, 11P21.

This work is supported by Natural Science Foundation of Shandong Province (Grant No. Q98A02110).

[1]

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The authors thank Professor W. G. Nowak for kindly sending reprints of some of his papers.

2. Some preliminary lemmas and results. The following lemmas are needed.

Lemma 1. Suppose 0 < c

₁

λ

₁

≤ |f

⁰

(n)| ≤ c

₂

λ

₁

and |f

⁰⁰

(n)| ∼ λ

₁

N

⁻¹

for N ≤ n ≤ cN . Then

X

N <n≤cN

e(f (n)) λ

⁻¹₁

+ λ

^1/2₁

N

^1/2

.

P r o o f. If c

2

λ

1

≤ 1/2, this estimate is contained in the Kuz’min–Landau inequality; otherwise, the estimate follows from the well-known van der Cor- put’s estimate for the second order derivative.

Lemma 2. Suppose a(n) = O(1), 0 < L ≤ M < N ≤ cL, L 1, T ≥ 2.

Then

X

M <n≤N

a(n) = 1 2πi

T

\

−T

X

L<l≤cL

a(l)

l

^it

· N

^it

− M

^it

t dt

+ O

min

1, L

T kM k

+ min

1, L

T kN k

+ O

L log(1 + L) T

. P r o o f. This is the well-known Perron formula.

Lemma 3. Suppose f (n) P and f

⁰

(n) ∆ for n ∼ N. Then X

n∼N

min

D, 1 kf (n)k

(P + 1)(D + ∆

⁻¹

) log(2 + ∆

⁻¹

).

P r o o f. This is contained in Lemma 2.8 of Kr¨atzel [9].

Lemma 4. Suppose a(n) are any complex numbers and 1 ≤ Q ≤ N. Then

X

N <n≤cN

a(n)

²

N

Q X

0≤q≤Q

1 − q

Q

< X

N <n≤cN −q

a(n)a(n + q).

P r o o f. This is Weyl’s inequality.

Lemma 5. Let M ≤ N < N

₁

≤ M

₁

and a(n) be any complex numbers.

Then

X

N <n≤N₁

a(n) ≤

∞

\

−∞

K(θ)

X

M <n≤M₁

a(m)e(mθ)

dθ

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with K(θ) = min(M

1

− M + 1, (π|θ|)

⁻¹

, (π|θ|)

⁻²

) and

∞

\

−∞

K(θ) dθ ≤ 3 log(2 + M

1

− M ).

P r o o f. This is Lemma 2.2 of Bombieri and Iwaniec [2].

Lemma 6. Let αβ 6= 0, ∆ > 0, M ≥ 1, N ≥ 1. Let A(M, N, ∆) be the number of quadruples (m, e m, n, e n) such that

|( e m/m)

^α

− (e n/n)

^β

| < ∆ with M ≤ m, e m ≤ 2M and N ≤ n, e n ≤ 2N . Then

A(M, N, ∆) M N log 2M N + ∆M

²

N

²

. P r o o f. This is Lemma 1 of [3].

Lemma 7. Suppose f (x) and g(x) are algebraic functions in [a, b] and

|f

⁰⁰

(x)| ∼ 1

R , |f

⁰⁰⁰

(x)| 1 RU ,

|g(x)| G, |g

⁰

(x)| GU

₁⁻¹

, U, U

1

≥ 1.

Then X

a<n≤b

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

α<u≤β

b

_u

g(n

_u

)

p |f

⁰⁰

(n

_u

)| e(f (n

_u

) − un

_u

+ 1/8)

+ O(G log(β − α + 2) + G(b − a + R)(U

⁻¹

+ U

₁⁻¹

)) + 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

⁰

(x), n

u

is the solution of the equation f

⁰

(x) = u,

b

_u

=

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

_n∈Z

{|t − n|}.

P r o o f. This is Theorem 2.2 of Min [10].

Lemma 8. Suppose A

_i

, B

_j

, a

_i

and b

_j

are all positive numbers. If Q

₁

and Q

2

are real with 0 < Q

1

≤ Q

2

, then there exists some q such that

Q

₁

≤ q ≤ Q

₂

and

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X

m i=1

A

_i

q

^aⁱ

+ X

n j=1

B

_j

q

^−b^j

≤ 2

^m+n

X

^m

i=1

X

n j=1

(A

^b_i^j

B

_j^aⁱ

)

^1/(aⁱ^+b^j⁾

+ X

m i=1

A

_i

Q

^a₁ⁱ

+ X

n j=1

B

_j

Q

^−b₂ ^j

.

P r o o f. This is Lemma 3 of Srinivasan [12].

Lemma 9. Suppose x is a large positive real number , M, N, U are positive integers such that 1 ≤ N ≤ M

^1/4

, x

^ε

≤ M ≤ x

^4/15

and M

^1/6

≤ U ≤ M

^5/6

. Then

S = X

n∼N

a

_n

X

u∼U

b

_u

X

v∼M U⁻¹

c

_v

e

√ nx uv

(2.1)

(x

^1/12

M

^7/12

N

^11/12

+ M

^11/12

N

^1/2

) log x,

where a

_n

, b

_u

, c

_v

are coefficients satisfying |a

_n

| ≤ 1, |b

_u

| ≤ 1, |c

_v

| ≤ 1.

P r o o f. We use Heath-Brown’s method [4].

Without loss of generality, suppose M

^1/2

≤ U ≤ M

^5/6

; otherwise we change the order of U and V = M U

⁻¹

. Suppose 1 ≤ Q ≤ V N is a parameter to be determined. For each q (1 ≤ q ≤ V N ) define

w

_q

=

(v, n)

v ∼ V, n ∼ N, 2 √

N (q − 1) V Q <

√ n v < 2 √

N q V Q

. Then

S = X

u∼U

b

u

X

Q q=1

X

(v,n)∈w_q

a

n

c

v

e

√ nx uv

. By Cauchy’s inequality

|S|

²

U Q X

u∼U

X

Q q=1

X

(v1,n1),(v2,n2)∈wq

a

_n₁

a

_n₂

(2.2)

× c

_v₁

c

_v₂

e

√ x u

√ n

1

v

₁

−

√ n

2

v

₂

U Q X

(∗)

X

u∼U

e

√ x u

√ n

₁

v

1

−

√ n

₂

v

2

, where (∗) denotes the condition

(2.3)

√ n

₁

v

1

−

√ n

₂

v

2

≤ 2 √

N

V Q , v

₁

, v

₂

∼ V, n

₁

, n

₂

∼ N.

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Let λ = √

n

1

/v

1

− √

n

2

/v

2

. Then by Lemma 1, the inner sum in (2.2) can be estimated by

(2.4) min

U, U

²

√ x|λ|

+ x

^1/4

|λ|

^1/2

U

^−1/2

.

By Lemma 6, the contribution of U (namely |λ| ≤ U x

^−1/2

) to |S|

²

is (2.5) U

²

Q(V N log 2V N + U x

^−1/2

N

^3/2

V

³

) U

²

QV N log 2V N.

The contribution of U

²

x

^−1/2

|λ|

⁻¹

(|λ| > U x

^−1/2

) is U

²

QV N (log 2V N )

²

by a similar argument. The contribution of x

^1/4

|λ|

^1/2

U

^−1/2

is (note |λ|

√ N /(V Q), Q V N )

x

^1/4

U

^1/2

N

^9/4

V

^3/2

Q

^−1/2

. Now, taking Q = 1 + [x

^1/6

U

⁻¹

N

^5/6

V

^1/3

], we get

(2.6) |S| log

⁻¹

2V N U V

^1/2

N

^1/2

+ x

^1/12

U

^1/2

V

^2/3

N

^11/12

, whence the lemma follows since M

^1/2

≤ U ≤ M

^5/6

.

Lemma 10. Suppose x, M, N satisfy the conditions of Lemma 9. Then T (M, N ) = X

n∼N

a(n) X

m∼M

µ(m)e

√ nx m

(2.7)

(x

^1/12

M

^7/12

N

^11/12

+ M

^11/12

N

^1/2

)x

^ε

, where |a(n)| ≤ 1.

P r o o f. By Heath-Brown’s identity (k = 4), T (M, N ) can be written as O(log

⁸

x) sums of the form

(2.8) T = X

n∼N

a(n) X

m₁∼M₁

. . . X

m₈∼M₈

µ(m

1

) . . . µ(m

4

)e

√

nx m

1

. . . m

8

, where M M

₁

. . . M

₈

M, M

₁

, M

₂

, M

₃

, M

₄

≤ (2M )

^1/4

. Some m

_i

may only take value 1.

To prove the lemma we consider three cases.

Case 1: There is some M

i

such that M

i

> M

^5/6

. It must follow that i ≥ 5; i = 8 for example. We use the exponent pair (1/6, 4/6) to estimate the sum on m

₈

and estimate other variables trivially, to get (F = √

N xM

⁻¹

) T N M M

₈⁻¹

M

₈

F +

F M

8

_1/6

M

₈^2/3

(2.9)

x

^−1/2

M

²

N

^1/2

+ x

^1/12

M

^5/12

N

^13/12

x

^1/12

M

^5/12

N

^13/12

x

^1/12

M

^7/12

N

^11/12

.

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Case 2: There is some M

i

satisfying M

^1/6

≤ M

i

≤ M

^5/6

. Let u = m

i

, v = Q

j6=i

m

_j

, U = M

_i

. Then by Lemma 9 we get

(2.10) x

^−ε

T x

^1/12

M

^7/12

N

^11/12

+ M

^11/12

N

^1/2

, where x

^ε

comes from the divisor argument.

Case 3: All M

_i

satisfy M

_i

≤ M

^1/6

. Without loss of generality, suppose M

1

≥ M

2

≥ . . . ≥ M

8

. Let l be the first positive integer j such that

(2.11) M

₁

. . . M

_j

> M

^1/6

; then l ≥ 2. Thus we have

M

₁

. . . M

_l

≤ (M

₁

. . . M

_l−1

)M

_l

≤ M

^2/6

.

Now take u = m

1

. . . m

j

, v = m

j+1

. . . m

6

, U = M

1

. . . M

j

. By Lemma 9, (2.10) still holds.

Lemma 10 follows from the three cases.

Now we prove the following proposition. It plays an important role in this paper. The idea of the proof has been used by several authors; see Jia [8], Baker and Harman [1], for example.

Proposition 1. Suppose M, N are large positive numbers, A > 0, α, β are rational numbers (not non-negative numbers). Suppose m ∼ M, n ∼ N, F = AM

^α

N

^β

N, |a(m)| ≤ 1, |b(n)| ≤ 1. Then

S = X

M <m≤2M

X

N <n≤2N

a(m)b(n)e(Am

^α

n

^β

)

(M N

^1/2

+ F

^4/20

M

^13/20

N

^15/20

+ F

^4/23

M

^15/23

N

^18/23

+ F

^1/6

M

^2/3

N

^7/9

+ F

^1/5

M

^3/5

N

^4/5

+ F

^1/10

M

^4/5

N

^7/10

) log

⁴

F. P r o o f. Without loss of generality, we suppose β > 0; for β < 0, the proof is the same. By Cauchy’s inequality and Lemma 4 we get

|S|

²

M X

M <m≤2M

X

N <n≤2N

b(n)e(Am

^α

n

^β

)

²

(2.12)

M

²

N

²

Q + M N Q |Σ

₁

| with

Σ

₁

= X

Q q=1

1 − q

Q

X

N <n≤2N −q

b(n)b(n + q) X

M <m≤2M

e(Am

^α

g(n, q)),

where Q is a parameter satisfying log N ≤ Q ≤ N log

⁻¹

N, g(n, q) =

(n + q)

^β

− n

^β

.

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We write

Σ

₁

= X

1≤q≤Q

X

2N −Q<n≤2N −q

X

m

+ X

q≤B

X

N <n≤2N −Q

X

m

(2.13)

+ X

B≤q≤Q

X

N <n≤2N −Q

X

m

= Σ

2

+ Σ

3

+ Σ

4

, where B = max(log N, M N /(c(α, β))F ) and

c(α, β) = 2|αβ| max(2

^α−1

, 1) max(2

^β−1

, 1).

By Lemma 1 we have Σ

₂

M N Q

F + F

^1/2

Q

^5/2

N

^1/2

log N, (2.14)

Σ

₃

M N

²

F + F

^1/2

N

^1/2

log

²

N. (2.15)

So we only need to bound Σ

₄

. Notice that Σ

₄

can be written as the sum of O(log Q) exponential sums of the form

(2.16) Σ

5

= X

Q1<q≤2Q1

c(q) X

N <n≤2N −Q

b(n)b(n+q) X

M <m≤2M

e(Am

^α

g(n, q)), where B ≤ Q

₁

≤ Q/2, c(q) = 1 − q/Q.

By Lemma 7 we have

(2.17) X

M <m≤2M

e(Am

^α

g(n, q))

= c

₃

X

U1<u≤U2

(Ag)

^{1/(2(1−α))}

u

(1−α/2)/(1−α)

e(c

₄

A

^1/(1−α)

g

^1/(1−α)

u

^{−α/(1−α)}

) + O

log F + M N

F q + min

M N

^1/2

F

^1/2

q

^1/2

, max

1 hU

2

i , 1

hU

1

i

, where U

₁

= c

₅

AM

^α−1

g, U

₂

= c

₆

AM

^α−1

g, g = g(n, q). By Lemma 3, the contribution of the error term to Σ

5

is

N Q

₁

log F + M N

²

F

⁻¹

log Q

₁

+ F

^1/2

Q

^3/2₁

N

^−1/2

.

It can be easily seen that g(n, q) < βqn

^β−1

for 0 < β < 1, g(n, q) >

βqn

^β−1

for β > 1. If βqAM

^α−1

n

^β−1

− AM

^α−1

g > log

⁻¹

N, then by the trivial estimate we have (i = 5, 6)

(2.18) X

q∼q₁

X

n∼N

X

c_iAM^α−1g<u≤c_iβqAM^α−1n^β−1

(Ag)

^{1/(2(1−α))}

u

(1−α/2)/(1−α)

× e(c

₄

A

^1/(1−α)

g

^1/(1−α)

u

^{−α/(1−α)}

)

F

^1/2

Q

^5/2₁

N

^−1/2

log

²

N.

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Now suppose βqAM

^α−1

n

^β−1

− AM

^α−1

g ≤ log

⁻¹

N. Notice that the fact that

[c

_i

βqAM

^α−1

n

^β−1

] − [c

_i

AM

^α−1

g] = 1 always implies

kc

_i

AM

^α−1

gk ≤ c

_i

βqAM

^α−1

n

^β−1

− c

_i

AM

^α−1

g = δ

_i

. We have

(2.19) X

c_iAM^α−1g<u≤c_iβqAM^α−1n^β−1

(Ag)

^{1/(2(1−α))}

u

(1−α/2)/(1−α)

M N

^1/2

F

^1/2

Q

^1/2₁

X

ciAM^α−1g<u≤ciβqAM^α−1n^β−1

1 M N

^1/2

F

^1/2

Q

^1/2₁

min

1, δ

_i

kc

i

AM

^α−1

gk

M N

^1/2

F

^1/2

Q

^1/2₁

min

1, F Q

²₁

M N

²

· 1 kc

_i

AM

^α−1

gk

,

thus by Lemma 3, (2.20) X

q∼q1

X

n∼N

X

ciAM^α−1g<u≤ciβqAM^α−1n^β−1

(Ag)

^{1/(2(1−α))}

u

(1−α/2)/(1−α)

× e(c

4

A

^1/(1−α)

g

^1/(1−α)

u

^{−α/(1−α)}

)

F

^1/2

Q

^3/2₁

N

^−1/2

log

²

N. Now it suffices to bound

Σ

6

= X

Q₁<q≤2Q₁

c(q) X

N <n≤2N −Q

b(n)b(n + q) (2.21)

× X

u

(Ag)

^{1/(2(1−α))}

u

(1−α/2)/(1−α)

e(c

₄

A

^1/(1−α)

g

^1/(1−α)

u

^{−α/(1−α)}

), where

c

₅

βqAM

^α−1

n

^β−1

< u ≤ c

₆

βqAM

^α−1

n

^β−1

. By Lemmas 2 and 3 we get (choose T = F

¹⁰

)

(2.22) Σ

₆

|Σ

₇

| + F

^1/2

Q

^3/2₁

N

^−1/2

log N

with

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Σ

₇

= X

Q1<q≤2Q1

c(q) X

N <n≤2N −Q

b(n)b(n + q) X

u

(Ag)

^{1/(2(1−α))}

u

(1−α/2)/(1−α)

(2.23)

× q

^it

A

^it

M

^(α−1)it

n

^(β−1)it

u

^it

e(c

4

A

^1/(1−α)

g

^1/(1−α)

u

^{−α/(1−α)}

), where t is a real number independent of the variables and

c

7

F Q

1

(M N )

⁻¹

< u ≤ c

8

F Q

1

(M N )

⁻¹

. It is easy to see that

Σ

7

= X

u

A

1/(2(1−α))+it

M

^(α−1)it

u

(1−α/2)/(1−α)+it

X

N <n≤2N −Q

b(n)n

^(β−1)it

(2.24)

× X

Q1<q≤2Q1

j(q)b(n + q)g

^{1/(2(1−α))}

× e(c

₄

A

^1/(1−α)

g

^1/(1−α)

u

^{−α/(1−α)}

)

X

u

(Ag)

^{1/(2(1−α))}

u

(1−α/2)/(1−α)

(Q

₁

N

^β−1

)

^{1/(2(1−α))}

Σ

₈

(u)

F

^1/2

Q

^1/2₁

N

^−1/2

Σ

₈

(u

₀

) for some u

₀

with

(2.25) Σ

₈

(u) = X

N <n≤2N −Q

X

Q₁<q≤2Q₁

j(q)b(n + q)g

₀^{1/(2(1−α))}

× e(c

₄

A

^1/(1−α)

g

^1/(1−α)

u

^{−α/(1−α)}

) , where j(q) = c(q)q

^it

, 1 g

0

= g(Q

1

N

^β−1

)

⁻¹

1. Suppose 10 ≤ R ≤ Q

1

log

^−1/2

N is a parameter to be determined. By Cauchy’s inequality and Lemma 4 we get

Σ

₈

(u)

²

N X

n

X

q

j(q)b(n + q)g

^{1/(2(1−α))}₀

(2.26)

× e(c

₄

A

^1/(1−α)

g

^1/(1−α)

u

^{−α/(1−α)}

)

²

N

²

Q

²₁

R + N Q

₁

R

X

1≤r≤R

|E

_r

|, where

E

r

= X

N <n≤2N −Q

X

Q₁<q≤2Q₁−r

j(q)b(n + q)g

₀^{1/(2(1−α))}

(n, q)

× j(q + r)b(n + q + r)g

^{1/(2(1−α))}₀

(n, q + r)

× e(c

₄

A

^1/(1−α)

u

^{−α/(1−α)}

(g

^{1/(2(1−α))}

(n, q) − g

^{1/(2(1−α))}

(n, q + r))).

(10)

So it reduces to bound E

r

for fixed r. Making the change of variable n + q = l, we have

E

_r

= X

Q1<q≤2Q1−r

j(q)j(q + r) X

N +q<l≤2N +q−Q

b(l)b(l + r)g

₀^{1/(2(1−α))}

(l − q, q)

× g

₀^{1/(2(1−α))}

(l − q, q + r)

× e(c

₄

A

^1/(1−α)

u

^{−α/(1−α)}

(g

^{1/(2(1−α))}

(l, q) − g

^{1/(2(1−α))}

(l, q + r)))

= X

N +Q1<n≤2N +2Q1−r−Q

b(n)b(n + r)

× X

max(Q1,n−2N +Q)<q≤min(2Q1−r,n−N )

j(q)j(q + r)

× g

₀^{1/(2(1−α))}

(n − q, q)g

₀^{1/(2(1−α))}

(n − q, q + r)

× e(c

₄

A

^1/(1−α)

u

^{−α/(1−α)}

k(n, q, r)), where

k(n, q, r) = g

^{1/(2(1−α))}

(n − q, q) − g

^{1/(2(1−α))}

(n − q, q + r).

By Lemma 5 we get E

_r

log

⁻¹

N X

n∼N

X

q∼Q₁

j(q)j(q + r)g

₀^{1/(2(1−α))}

(n − q, q)

× g

^{1/(2(1−α))}₀

(n − q, q + r)e(c

₄

A

^1/(1−α)

u

^{−α/(1−α)}

k(n, q, r) + θ

₀

q) , where θ

₀

is a real number independent of n and q.

Suppose 10 ≤ T ≤ Q

₁

log

^−1/2

N is a parameter to be determined. By Cauchy’s inequality and Lemma 4 we get

(2.27) |E

_r

|

²

log

⁻²

N N

²

Q

²₁

T + N Q

1

T |D

_t

(r)|, with

D

t

(r) = X

n∼N

X

Q1<q≤2Q1−t

j(q)j(q + r)g

^{1/(2(1−α))}₀

(n − q, q)

× g

₀^{1/(2(1−α))}

(n − q, q + r)j(q + t + r)j(q + t)

× g

₀^{1/(2(1−α))}

(n − q − t, q + t)g

₀^{1/(2(1−α))}

(n − q − t, q + t + r)

× e(c

₄

A

^1/(1−α)

u

^{−α/(1−α)}

(k(n, q, r) − k(n, q + t, r)))

X

q∼Q₁

X

n∼N

φ(n)e(f (n))

,

(11)

where

φ(n) = g

₀^{1/(2(1−α))}

(n − q, q)g

^{1/(2(1−α))}₀

(n − q, q + r)

× g

₀^{1/(2(1−α))}

(n − q − t, q + t)g

^{1/(2(1−α))}₀

(n − q − t, q + t + r) and

f (n) = c

₄

A

^1/(1−α)

u

^{−α/(1−α)}

(k(n, q, r) − k(n, q + t, r)).

It is an easy exercise to verify that φ(n) is monotonic and

|f

⁰

(n)| ∼ F rt

Q

₁

N

²

, |f

⁰⁰

(n)| ∼ F rt

Q

₁

N

³

(N < n ≤ 2N );

thus by Lemma 1 we get (2.28) D

_t

(r) Q

₁

Q

₁

N

²

F rt + (F rt)

^1/2

(Q

1

N )

^1/2

= Q

²₁

N

²

F rt + (F rtQ

₁

)

^1/2

N

^1/2

. Inserting (2.28) into (2.27) we get

(2.29) |E

_r

|

²

log

⁻²

N Q

²₁

N

²

T + Q

³₁

N

³

F rT + Q

^3/2₁

(N F T r)

^1/2

.

Notice that (2.29) is also true for 0 < T < 10; then by Lemma 8 choosing a best T ∈ (0, Q

₁

log

^−1/2

N ) we get

(2.30) |E

_r

|

²

log

⁻²

N N

^1/2

Q

^5/6₁

F

^1/6

r

^1/6

+ N

^2/3

Q

₁

+ N Q

^1/2₁

+ N

^3/2

Q

₁

F

^1/2

r

^1/2

. Inserting (2.30) into (2.26) we have

(2.31) Σ

8

(u)

²

log

⁻²

N

Q

²₁

N

²

R + N

^3/2

Q

^11/6₁

F

^1/6

R

^1/6

+ N

^5/2

Q

²₁

F

^1/2

R

^1/2

+ N

^5/3

Q

²₁

+ N

²

Q

^3/2₁

. This is also true for 0 < R ≤ 10. Choosing a best R ∈ (0, Q

₁

log

^−1/2

N ) via Lemma 8 we get

Σ

8

(u) log

⁻²

N N

^11/14

Q

^13/14₁

F

^1/14

+ N

^14/16

Q

^15/16₁

(2.32)

+ N

^5/4

Q

^3/4₁

F

^−1/4

+ N

^5/6

Q

1

+ N Q

^3/4₁

. Inserting (2.32) into (2.24) we have

Σ

₇

N

^4/14

Q

^20/14₁

F

^8/14

+ N

^6/16

Q

^23/16₁

F

^8/16

+ N

^3/4

Q

^5/4₁

F

^1/4

(2.33)

+ N

^1/3

Q

^3/2₁

F

^1/2

+ N

^1/2

Q

^5/4₁

F

^1/2

.

Combining (2.17), (2.18), (2.20), (2.22) and (2.33) we get Σ

5

log

⁻⁴

F N

^4/14

Q

^20/14

F

^8/14

+ N

^6/16

Q

^23/16

F

^8/16

(2.34)

+ N

^3/4

Q

^5/4

F

^1/4

+ N

^1/3

Q

^3/2

F

^1/2

+ N

^1/2

Q

^5/4

F

^1/2

+ N Q + M N

²

F

⁻¹

+ F

^1/2

Q

^5/2

N

^−1/2

.

(12)

Inserting (2.14), (2.15) and (2.34) into (2.12) we get

|S|

²

log

⁻⁶

F M

²

N

²

Q + M N

^18/14

Q

^6/14

F

^8/14

(2.35)

+ M N

^22/16

Q

^7/16

F

^8/16

+ M N

^7/4

Q

^1/4

F

^1/4

+ M N

^4/3

Q

^1/2

F

^1/2

+ M N

^3/2

Q

^1/4

F

^1/2

+ M N

²

+ M

²

N

³

Q

⁻¹

F

⁻¹

+ M N

^1/2

Q

^3/2

F

^1/2

.

Since Q < N F, we have

M N

²

+ M N

^7/4

Q

^1/4

F

^1/4

M N

^3/2

Q

^1/4

F

^1/2

, M

²

N

³

(F Q)

⁻¹

M

²

N

²

Q

⁻¹

. So we obtain

|S|

²

log

⁻⁶

F M

²

N

²

Q + M N

^18/14

Q

^6/14

F

^8/14

(2.36)

+ M N

^22/16

Q

^7/16

F

^8/16

+ M N

^4/3

Q

^1/2

F

^1/2

+ M N

^3/2

Q

^1/4

F

^1/2

+ M N

^1/2

Q

^3/2

F

^1/2

.

Note that (2.36) is trivial for 0 < Q ≤ log N. Now the proposition follows from choosing a best Q ∈ (0, N log

⁻¹

N ) via Lemma 8.

3. An expression of the error term. In the rest of this paper, we always use E

P

(x) to denote the difference V (x)−

_π⁶

x. The aim of this section is to give an expression of E

_P

(x) subject to RH.

Let y be a parameter, x

^ε

≤ y ≤ x

^1/2−ε

, (3.1) f

1

(s) = X

n≤y

µ(n)n

^−s

, f

2

(s) = ζ

⁻¹

(s) − f

1

(s).

Let r(n) be the number of representations of n as a sum of two squares.

Then

V (x) = X

a²+b²≤x (a,b)=1

1 = X

a²+b²≤x

X

m|(a,b)

µ(m) (3.2)

= X

m²(a²+b²)≤x

µ(m) = X

m²k≤x

µ(m)r(k)

= X

m≤y

+ X

m>y

= Σ

₁

+ Σ

₂

, say.

Notice that for σ > 1, (3.3)

X

∞ n=1

r(n)n

^−s

= 4ζ(s)L(s, χ),

(13)

where χ is the non-principal character mod 4, we have Σ

₁

= X

m≤y

µ(m)P

x m

²

(3.4)

= X

m≤y

µ(m)

Res

s=1

4ζ(s)L(s, χ) s

x m

²

_s

+ E

x m

²

= Res

_s=1

(4f

₁

(2s)ζ(s)L(s, χ)x

^s

s

⁻¹

) + X

m≤y

µ(m)E

x m

²

. To treat Σ

₂

we begin with

f

2

(s) = X

m>y

µ(m)m

^−s

for σ > 1. Hence

(3.5) 4f

₂

(2s)ζ(s)L(s, χ) = X

∞ n=1

b(n)n

^−s

(σ > 1), where

b(n) = X

n=m²k, m>y

µ(m)r(k).

By Perron’s formula we have

(3.6) Σ

₂

= X

n≤x

b(n) = 1 2πi

1+ε+ix

\

²

1+ε−ix²

g(s) ds + O(x

^ε

), where

g(s) = 4f

₂

(2s)ζ(s)L(s, χ)x

^s

s

⁻¹

. By Cauchy’s theorem, we have

(3.7) 1

2πi

1+ε+ix

\

²

1+ε−ix²

g(s) ds = I

1

+ I

2

− I

3

+ Res

s=1

g(s), where

I

1

= 1 2πi

1+ε+ix

\

²

0.5+ε+ix²

g(s) ds, I

2

= 1 2πi

0.5+ε+ix

\

²

0.5+ε−ix²

g(s) ds,

I

₃

= 1 2πi

1+ε−ix

\

²

0.5+ε−ix²

g(s) ds.

Since RH is true, it follows that

ζ(s) |t|

^ε

+ 1, σ ≥ 0.5 + ε,

(3.8)

(14)

f

2

(2s) y

^−1/2

(|t|

^ε

+ 1), σ ≥ 0.5 + ε.

(3.9)

For L(s, χ), we have

(3.10) L(s, χ) (|t| + 1)

^(1−σ)/2

, 0.5 ≤ σ ≤ 1.

Using (3.8)–(3.10) we get

I

₁

− I

₃

y

^−1/2

(1 + x

^ε

), (3.11)

I

₂

y

^−1/2

x

^1/2+ε

_x

\

²

1

|L(1/2 + ε + it, χ)|

t dt + 1

(3.12)

y

^−1/2

x

^1/2+ε

. Combining (3.6)–(3.12) we get

(3.13) Σ

2

= Res

s=1

(4f

2

(2s)ζ(s)L(s, χ)x

^s

s

⁻¹

) + O(y

^−1/2

x

^1/2+ε

).

From (3.2), (3.4) and (3.13) we get

V (x) = Res

_s=1

(4ζ

⁻¹

(2s)ζ(s)L(s, χ)x

^s

s

⁻¹

) (3.14)

+ X

m≤y

µ(m)E

x m

²

+ O(y

^−1/2

x

^1/2+ε

)

= 6

π x + X

m≤y

µ(m)E

x m

²

+ O(y

^−1/2

x

^1/2+ε

).

Now we obtain the main result of this section.

Proposition 2. If RH is true, then for x

^ε

≤ y ≤ x

^1/2−ε

, we have (3.15) E

_P

(x) = X

m≤y

µ(m)E

x m

²

+ O(y

^−1/2

x

^1/2+ε

).

4. Proof of the Theorem. Take y = x

^4/15

in Proposition 2. We only need to estimate the sum

X

m∼M

µ(m)E

x m

²

for x

^1/10

M y. For M x

^1/10

, we have trivially

(4.1) X

m≤x^1/10

µ(m)E

x m

²

X

m≤x^1/10

x

^1/3

m

^−2/3

x

^11/30

.

By the well-known Voronoi formula of E(t) (see [7], 13.8) we get

(15)

(4.2) x

^−ε

X

m∼M

µ(m)E

x m

²

X

m∼M

µ(m)x

^1/4

m

^1/2

X

n≤x^4/15

r(n) n

^3/4

e

√ nx m

+ x

^11/30

. It now suffices to show that (1 N Y )

(4.3) S(M, N ) = X

m∼M

µ(m)x

^1/4

m

^1/2

X

n∼N

r(n) n

^3/4

e

√ nx m

x

^11/30+ε

. We consider three cases.

Case 1: M ≤ N ≤ x

^4/15

. Let T (M, N ) = X

m∼M

µ(m)M

^1/2

m

^1/2

X

n∼N

r(n)N

^3/4−ε

n

^3/4

e

√ nx m

. By Proposition 1 (take (X, Y ) = (N, M )) we get

x

^−ε

T (M, N ) N M

^1/2

+ x

^2/20

N

^15/20

M

^11/20

(4.4)

+ x

^2/23

N

^17/23

M

^14/23

+ x

^1/12

N

^15/20

M

^11/18

+ x

^2/20

N

^14/20

M

^12/20

+ x

^1/20

N

^17/20

M

^12/20

, whence

x

^−ε

S(M, N ) x

^1/4

N

^1/4

+ x

^7/20

M

^1/20

+ x

^31/92

M

^9/92

(4.5)

+ x

^1/3

M

^1/9

+ x

^3/10

N

^1/10

M

^1/10

x

^109/300

x

^11/30

.

Case 2: M

^1/4

≤ N ≤ M . We again use Proposition 1 to bound S(M, N ) (now take (X, Y ) = (M, N )) and get

x

^−ε

S(M, N ) x

^7/20

N

^1/10

M

^−1/20

+ x

^31/92

N

^11/92

M

^−1/46

(4.6)

+ x

^1/3

N

^1/9

+ x

^7/20

N

^3/20

M

^−1/10

+ x

^3/10

M

^1/5

+ x

^1/4

N

^−1/4

M

^1/2

x

^109/300

+ x

^1/4

M

^7/16

x

^11/30

.

Case 3: N < M

^1/4

. We use Lemma 10 to bound T (M, N ) and get x

^−ε

T (M, N ) x

^1/12

N

^11/12

M

^1/12

+ N

^1/2

M

^11/12

,

whence we have

x

^−ε

S(M, N ) x

^1/3

N

^1/6

M

^1/12

+ x

^1/4

M

^5/12

(4.7)

+ x

^1/3

M

^1/8

+ x

^1/4

M

^5/12

x

^11/30

.

This completes the proof of the Theorem.

(16)

Acknowledgements. The authors thank the referee for his or her kind and 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. B o m b i e r i and H. I w a n i e c, On the order of ζ(

¹₂

On the number of coprime integer pairs within a circle

XC.1 (1999)