REPRESENTATION OF NATURAL NUMBERS BY SUMS OF FOUR. SQUARES OF INTEGERS HAVING A SPECIAL FORM. S. A. Gritsenko and N. N. Mot'kina.
Journal of Mathematical Sciences, Vol. 173, No. 2, 2011
REPRESENTATION OF NATURAL NUMBERS BY SUMS OF FOUR SQUARES OF INTEGERS HAVING A SPECIAL FORM UDC 511.51
S. A. Gritsenko and N. N. Mot’kina
Abstract. This paper obtains an asymptotic formula for the number of solutions to the equation l12 + l22 + l32 + l42 = N in integers l1 , l2 , l3 , l4 such that a < {ηlj } < b, where η is a quadratic irrational number, 0 ≤ a < b ≤ 1, j = 1, 2, 3, 4.
CONTENTS 1. Introduction . . . . . . 2. Auxiliary Assertions . 3. Proof of Theorem 1.1 References . . . . . . .
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194 195 195 200
Introduction
In 1770, Lagrange proved that each natural number N can be represented as the sum of four squares of integers: l12 + l22 + l32 + l42 = N.
(1.1)
Denote by I(N ) the number of representations (1.1).The following asymptotic formula is known for I(N ) (see [4]): 1 4 −2πiN a/q Sq,a e + O(N 17/18+ε ), I(N ) = π 2 N q4 1≤q
1≤a≤q (a,q)=1
where Sq,a =
q
e2πiaj
2 /q
j=1
is the Gaussian sum, and ε is an arbitrary positive number. Let us consider a subset of the set of integers having the form A = {l | a < {ηl} < b}, where η is a quadratic irrational number, and a and b are arbitrary numbers from the closed interval [0, 1]. Denote by J(N ) the number of solutions of Eq. (1.1) in integers from the special set A. Theorem 1.1. The following formula holds for any positive small ε: J(N ) = (b − a)4 I(N ) + O(N 0,9+3ε ). Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 67, Partial Differential Equations, 2010.
194
c 2011 Springer Science+Business Media, Inc. 1072–3374/11/1732–0194
2.
Auxiliary Assertions
Lemma 2.1 (see [5]). For any tuple a, b, c, d ∈ Z such that ad − bc = 1, ab and cd are even and for any ξ, ξ 8 = 1, the following relation holds: ar + b z πicz 2 1/2 ; = ξ(cr + d) exp θ(z, r), θ cr + d cr + d cr + d where
∞
θ(z, r) =
exp(πin2 r + 2πinz)
n=−∞
is an analytic function and z, r ∈ C, Im r > 0. Lemma 2.2 (see [1, 3]). The following assertions hold for the Gauss sum 2 S(q, a, b) = e2πi(aj +bj)/q , Sq,a = S(q, a, 0) : 1≤j≤q
(1) if (q, a) = d, then
S(q, a, b) =
(2) if (a, 2) = 1, then S2γ ,a
dS(q/d, a/d, b/d) 0
⎧ ⎪ ⎨0 = 2γ/2 (1 + iγ ) ⎪ ⎩ (γ+1)/2 2πiγ/8 e 2
(3) if (q, 2a) = 1, then S(q, a, b) = e−2πi4ab (4) if (q, 2) = 1, then
2 /q
a Sq,1 , q
√
q √ i q
Sq,1 =
if if if
if if
if if
d | b, d b;
γ = 1, γ is even, γ > 1 is odd;
4a4a ≡ 1
(mod q);
q ≡ 1 (mod 4), q ≡ −1 (mod 4).
Lemma 2.3 (see [2, 7]). K(q, a, b) τ (q)q 1/2 (a, b, q)1/2 ,
2πi(aj+b¯j)/q e is the Kloosterman sum, j ¯j ≡ 1 (mod q). where K(q, a, b) = 1≤j≤q (j,q)=1
3. 1. Let us extend the function ψ0 (x) =
Proof of Theorem 1.1
1 0
if if
a < x < b, 0 ≤ x ≤ a or
b≤x≤1
to the whole real line in a periodic way with period 1. Let Sk (β) =
∞
e2πiβl
2 −l2 /N
ψk (ηl),
k = 0, 1, 2;
l=−∞
195
then
1
S04 (β)e−2πiβN dβ.
J(N ) = e 0
By the I. M. Vinogradov “small glass” lemma (see [6, p. 22]), we choose r = [log N ], Δ = N 0,1 . For α = a + Δ/2 and β = b − Δ/2, the function ψ from this lemma is denoted by ψ1 . Setting α = a − Δ/2 and β = b + Δ/2, denote by ψ2 the corresponding function ψ. Then the following inequality hold: J1 (N ) ≤ J(N ) ≤ J2 (N ), where
1 Jk (N ) = e
Sk4 (β)e−2πiβN dβ,
k = 1, 2.
(3.1)
0
For J1 (N ) and J2 (N ), let us deduce asymptotic formulas whose principal terms are identical. Let us substitute the Fourier series expansion ck (m)e2πimηl + O(N − log π ) ψk (ηl) = |m|≤rΔ−1
of the function ψk (ηl) in (3.1): ck (m1 ) Jk (N ) = e |m1
1 ×
|≤rΔ−1
|m2
ck (m2 )
|≤rΔ−1
|m3
ck (m3 )
|≤rΔ−1
|m4
ck (m4 )
|≤rΔ−1
S(β, m1 )S(β, m2 )S(β, m3 )S(β, m4 )e−2πiβN dβ + O(N 2−log π ),
0
where S(β, m) =
∞
e2πiβl
2 −l2 /N +2πimηl
.
l=−∞
Note that for m1 = m2 = m3 = m4 = 0. the principal term Jk (N ) has the form c4k (0)I(N ). 2. In the sum S(β, m), for 0 < |m| ≤ r/Δ, let us set √ 1 d , β = + y1 , (d, q) = 1, 1 ≤ q ≤ τ, τ = [ N ], |y1 | < q qτ l = j + qn. Then 2 q dj i + mηj + πi 2y1 + j2 S(β, m) = exp 2πi q πN j=1 ∞ i i × exp πi 2y1 + q 2 n2 + 2πi mη + j 2y1 + qn . πN πN n=−∞ Let us make the change t = 2y1 + then we obtain S(β, m) =
q j=1
196
i , πN
e2πidj
2 /q
y = mη + jt; 2
e2πimηj eπitj θ(yq, tq 2 ).
From the functional equation for the theta-function (Lemma 2.1), let us express θ(yq, tq 2 ): 1 y 2 −1 2 −1/2 −πiy 2 /t ;− e θ θ(yq, tq ) = ξ (tq ) . tq tq 2 Then
∞ πi 1 2 exp − 2 (n − mηq) S(q, d, n), S(β, m) = √ tq ξq t n=−∞
where S(q, d, n) =
q j=1
dj 2 + nj exp 2πi q
is the Gaussian sum. 3. Let d /q , d/q, and d /q be neighboring Farey fractions such that d d d < < , q q q
τ < q + q,
q + q < τ + q.
For (m1 , m2 , m3 , m4 ) = (0, 0, 0, 0), let us consider I(N, m1 , m2 , m3 , m4 ) 1 =
S(β, m1 )S(β, m2 )S(β, m3 )S(β, m4 )e−2πiβN dβ
0 τ 1 1 = 4 ξ q4 q=1
)) 1/(q(q+q
e −1/(q(q+q ))
−2πiyN
1 t2
∞
∞
∞
n1 =−∞ n2 =−∞ n3 =−∞
πi × exp − 2 (n1 − m1 ηq)2 + (n2 − m2 ηq)2 + (n3 − m3 ηq)2 + (n4 − m4 ηq)2 tq n4 =−∞ S(q, d, n1 )S(q, d, n2 )S(q, d, n3 )S(q, d, n4 )e−2πidN/q dy. × ∞
1≤d≤q (d,q)=1
4. Knowing the exact values of the Gaussian sums and the A. Weil estimate for the Kloosterman sum, let us estimate S(q, d, n1 )S(q, d, n2 )S(q, d, n3 )S(q, d, n4 )e−2πidN/q W = 1≤d≤q (d,q)=1
q 5/2 τ (q)(N, q)1/2 . For this purpose, let us represent q in the form 2γ q1 , where (2, q1 ) = 1. If x1 , x2 vary in the complete systems of residues modulo 2γ and q1 , respectively, then the form q1 x1 + 2γ x2 varies in the complete system of residues modulo q. Then we have S(q, d, ni ) = S(2γ , dq1 , ni )S(q1 , d2γ , ni ). Furthermore, let d = q1 a1 + 2γ a2 , where a1 and a2 vary in the reduced systems of residues modulo 2γ and q1 , respectively. Then S(q, d, ni ) = S(2γ , a1 , ni q¯1 )S(q1 , a2 , ni ¯2γ ), 197
where q1 q¯1 ≡ 1 (mod 2γ ), 2¯ 2 ≡ 1 (mod q1 ). We obtain W = W1 × W2 , where S(2γ , a1 , n1 q¯1 )S(2γ , a1 , n2 q¯1 )S(2γ , a1 , n3 q¯1 ) W1 = 1≤a1 ≤2γ (a1 ,2γ )=1
a1 − 2πi γ N , 2
× S(2 , a1 , n4 q¯1 ) exp S(q1 , a2 , n1 ¯2γ )S(q1 , a2 , n2 ¯2γ )S(q1 , a2 , n3 ¯2γ ) W2 = γ
1≤a2 ≤q1 (a2 ,q1 )=1
× S(q1 , a2 , n4 2 ) exp ¯γ
a2 − 2πi N . q1
For odd ni q¯1 , let us partition the sum S(2γ , a1 , ni q¯1 ) into two sums with summands of the form j = 2x and j = 2x + 1. Then γ−1 2 2πi(2a1 x2 + ni q¯1 x) γ exp S(2 , a1 , ni q¯1 ) = 2γ−1 + exp
x=1
2πi(a1 + ni q¯1 ) 2γ
2γ−1 −1
exp
x=0
2πi(2a1 x2 + (2a1 + ni q¯1 )x) . 2γ−1
In this case, by Lemma 2.2 (1), S(2γ , a1 , ni q¯1 ) = 0. For even ni q¯1 , S(2γ , a1 , ni q¯1 ) 2γ a1 (x + a ¯1 ni q¯1 /2)2 a ¯1 (ni q¯1 /2)2 . exp 2πi = exp − 2πi 2γ 2γ x=1
By Lemma 2.2 (2), we have
⎛
⎜ ⎜ S(2 , a1 , ni q¯1 ) = exp ⎜−2πi ⎝ γ
a ¯1 n2i
⎛
Then 2γ+4
W1 ≤ 2
1≤a1 ≤2γ (a1 ,2γ )=1
q¯1 2
2γ
2 ⎞
⎧ ⎟ ⎪ ⎨0 ⎟ ⎟ × 2γ/2 (1 + iγ ) ⎠ ⎪ ⎩ (γ+1)/2 2πiγ/8 e 2
if if if
⎛
⎜ ⎜ −a1 N − ⎜ ⎜ exp ⎜2πi ⎜ ⎝ ⎝
a ¯1 (n21
+
n22
+
n23
+
n24 )
2γ
γ = 1, γ is even, γ > 1 is odd. q¯1 2
2 ⎞⎞ ⎟⎟ ⎟⎟ ⎟⎟ . ⎠⎠
Finally, by Lemma 2.3, we have W1 τ (2γ )25γ/2 (N, n21 + n22 + n23 + n24 , 2γ )1/2 . Applying Lemma 2.2 (3) and 2.2 (4) to the sums S(q1 , a2 , ni ¯2γ ), we obtain −a2 N − a ¯2 (n21 + n22 + n23 + n24 )¯22+2γ 2 . exp 2πi W 2 = q1 q1 1≤a2 ≤q1 (a2 ,q1 )=1
Then, by Lemma 2.3,
5/2
W2 τ (q1 )q1 (N, n21 + n22 + n23 + n24 , q1 )1/2 . As a result, we obtain the desired estimate for |W |. 198
5. Let us substitute the estimate for |W | in the integral I(N, m1 , m2 , m3 , m4 ): τ 1 ε q (N, q)1/2 I(N, m1 , m2 , m3 , m4 ) 3/2 q q=1
)) 1/(q(q+q
−1/(q(q+q ))
N 2 dy 1 + (yN )2
2 π N ((n1 − m1 ηq)2 + (n2 − m2 ηq)2 + (n3 − m3 ηq)2 + (n4 − m4 ηq)2 ) . × exp − q 2 (1 + (2πyN )2 ) |ni | 0 such that π 2 N mi ηq 2 2ε e−kN . exp − 2 2 q (1 + (2πyN ) ) For other values of ni and for an arbitrary positive k, we have π 2 N (ni − mi ηq)2 exp − 2 exp(−kN 0,6+2ε (ni − mi ηq)2 ). q (1 + (2πyN )2 ) 199
As a result, we obtain S2,1 e
1
1≤q≤N 0,2−ε
q 3/2
−kN 2ε
e
−kN 2ε
N 1,2
δ −2
1
1≤q≤N 0,2−ε
q 3/2
×
∞
e−k((n1 −m1 ηq)
n1 ,n2 ,n3 ,n4 =−∞
N 0,8+2ε
2 +(n
S2,3
N 0,2−ε
