Small prime solutions of a nonlinear equation

Accepted Manuscript Small prime solutions of a nonlinear equation

Zhixin Liu

PII: DOI: Reference:

S0022-314X(17)30355-4 https://doi.org/10.1016/j.jnt.2017.09.014 YJNTH 5884

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Journal of Number Theory

Received date: Accepted date:

18 April 2017 3 September 2017

Please cite this article in press as: Z. Liu, Small prime solutions of a nonlinear equation, J. Number Theory (2018), https://doi.org/10.1016/j.jnt.2017.09.014

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SMALL PRIME SOLUTIONS OF A NONLINEAR EQUATION ZHIXIN LIU Abstract. Let a1 , · · · , a4 be non-zero integers and n any integer. Suppose that a1 , · · · , a4 and n satisfy some related conditions. In this paper we prove that (i) if aj are not all of the same sign, then the equation a1 p1 + a2 p22 + a3 p23 + a4 p24 = n has prime solutions satisfying max{p1 , p22 , p23 , p24 }  |n|+max{|aj |}14+ε ; (ii) if all aj are positive and n  max{|aj |}15+ε , then the equation a1 p1 + a2 p22 + a3 p23 + a4 p24 = n is soluble in primes pj .

1. Introduction Let n be an integer, and let a1 , · · · , a4 be non-zero integers. We consider here the nonlinear equation in the form a1 p1 + a2 p22 + a3 p23 + a4 p24 = n,

(1.1)

where pj are prime variables. We shall assume the condition of the congruent solubility for (1.1), that is N (q) ≥ 1

(1.2)

for all q ≥ 1,

where N (q) := card{(n1 , · · · , n4 ) :1 ≤ nj ≤ q, (nj , q) = 1, a1 n1 + a2 n22 + a3 n23 + a4 n24 ≡ n(mod q}. We also suppose that (1.3)

(ai , aj ) = 1,

1 ≤ i < j ≤ 4,

and write A = max{2, |a1 |, · · · , |a4 |}. The main results in this paper are the following two parallel theorems. Theorem 1.1. Suppose (1.2) and (1.3). If a1 , · · · , a4 are not all of the same sign, then (1.1) has solutions in primes pj satisfying max{p1 , p22 , p23 , p24 }  |n| + A14+ε , where the implied constant depends only on ε. Theorem 1.2. Suppose (1.2) and (1.3). If a1 , · · · , a4 are all positive, then (1.1) is soluble whenever (1.4)

n  A15+ε ,

where the implied constant depends only on ε. 2010 Mathematics Subject Classification. 11P32, 11P05, 11P55. Key words and phrases. small prime, Waring-Goldbach problem, circle method. 1

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ZHIXIN LIU

Theorem 1.2 with a1 = a2 = a3 = a4 = 1 is a classical result of Hua [2] in 1938. The equation (1.1) in general was first studied by Li, Zhao and Wang [4], who obtained a qualitative bound AC , in the place of A14+ε and A15+ε in Theorems 1.1 and 1.2 above, without the explicit values of the constant C. However, in [4] the condition (1.3) is relaxed to that (a1 , · · · , a4 ) = 1. We prove our theorem by the circle method. At this stage, we point out that in contrast to [4], which treated the enlarged major arcs by the Deuring-Heilbronn phenomenon, we show that under the stronger condition (1.3), the possible existence of Siegel’s zero does not have special influence and hence the Deuring-Heilbronn phenomenon can be avoided. This observation enables us to get better results without using heavy numerical computations. To treat the estimation of integral of minor arcs, we also use a new idea coming from [7]. 2. Outline of the method We denote by R(n) the weighted number of solutions of (1.1), i.e.  R(n) = (log p1 ) · · · (log p4 ), 2 2 n = a 1 p1 + a 2 p2 2 + a 3 p3 + a 4 p4 M < |a1 |p1 ≤ N M < |aj |p2 j ≤ N, j = 2, 3, 4

where M = N/200. We will investigate R(n) by the circle method. Let M(Q) =



q 

1≤q≤Q

a=1 (a, q) = 1

M(Q; q, a),

Q a Q where M(Q; q, a) denotes the interval [ aq − qN , q + qN ]. We introduce the parameters

Q1 = (N/A)9/40−ε ,

Q2 = (N/A)5/14 .

Let M = M(Q1 ) and m = [0, 1) \ M. We define m1 = m \ M(Q2 ),

m2 = M(Q2 ) \ M(Q1 ).

We further define the exponential sum  S1 (a1 α) =

(log p)e(a1 pα),

M <|a1 |p≤N

and



S2 (aj α) =

(log p)e(aj p2 α)

for

j = 2, 3, 4.

M <|aj |p2 ≤N

By orthogonality, we have  1 R(n) = S1 (a1 α)S2 (a2 α)S2 (a3 α)S2 (a4 α)e(−nα)dα   0 + + . = M

m1

m2

We start our proof by giving Lemma 2.1, which deals with the integral on the major arcs M.

SMALL PRIME SOLUTIONS OF A NONLINEAR EQUATION

Lemma 2.1. We have  S1 (a1 α)S2 (a2 α)S2 (a3 α)S2 (a4 α)e(−nα)dα = S(n, Q1 )J(n)+O( M

3

N 3/2 ), |a1 ||a2 a3 a4 |1/2 L

where S(n, Q1 ) and J(n) are defined in (2.1) and (2.2) respectively. The lemma is now standard by the iterative method introduced by [3] (Theorem 2 in [3]). To derive Lemma 2.1, we need to bound S(n, Q1 ) and J(n) from below. For χ mod q, we define q  ah χ(h)e( ), C1 (q, a) = C1 (χ0 , a), C1 (χ, a) = q h=1

and C2 (χ, a) =

q 

χ(h)e(

h=1

ah2 ), q

C2 (q, a) = C2 (χ0 , a).

If χ1 , · · · , χ4 are characters mod q, then we write q  hn e(− )C1 (χ1 , a1 h)C2 (χ2 , a2 h)C2 (χ3 , a3 h)C2 (χ4 , a4 h), B(n, q, χ1 , χ2 , χ3 , χ4 ) = q h=1 (h,q)=1

B(n, q) = B(n, q, χ0 , · · · , χ0 ), and (2.1)

S(n, Q1 ) =



A(n, q) =

B(n, q) , ϕ4 (q)

A(n, q).

q≤Q1

Lemma 2.2. Assuming (1.2) and (1.3), we have S(n, Q1 )  (log log A)−c for some constant c > 0. Lemma 2.3. Suppose (1.3) and (i) a1 , · · · , a4 are not all of the same sign and N ≥ 8|n|; or (ii) a1 , · · · , a4 are positive and n = N . Then we have  N 3/2 (2.2) J(n) := (m2 m3 m4 )−1/2  . |a1 ||a2 a3 a4 |1/2 a1 m1 +···+a4 m4 =n M <|aj |mj ≤N

3. Proofs of Lemmas 2.2 and 2.3 A necessary condition for the solubility of the equation(1.1) is the congruence solubility that N (q) ≥ 1 for all integer q ≥ 1. It is known (see (2) and (3) of Lemma 6.5 in [4]) that N (q) is a multiplicative function of q and N (pt ) ≥ 1 if and only if N (p) ≥ 1 for odd prime p and t ≥ 1 and N (2t ) ≥ 1 if and only if N (8) ≥ 1 for t ≥ 3. Thus, actually, it is only needed that N (2), N (4), N (8) and N (p) are larger or equal to 1 for odd prime p in Theorems 1.1 and 1.2.

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ZHIXIN LIU

Proof of Lemma 2.2. Lemma 2.2 is a consequence of the following Lemma 3.2. In order to prove Lemma 3.2, we need Lemma 3.1. Lemma 3.1. Suppose (1.3). We have |A(n, p)|  p−1 . Moreover, |A(n, p)| < 60p−2 for all p  a1 · · · a4 . Proof. The second half of this lemma have been proved in Lemma 6.6 of [4]. All we left is the proof of the first half of this lemma. Among the numbers a2 , · · · a4 , −n, let m of them be divisible by p and k (respectively l) of them be quadratic residues (respectively non-residues) modulo p. Then from the proof of Lemma 6.5 in [4], we have pϕ(p)−4 N (p) = 1 + A(n, p),

(3.1) m + k + l = 4 and if p|a1 A(n, p) =

  1 ϕ(p)m−3 (λ − 1)k (−λ − 1)l + (λ − 1)l (−λ − 1)k , 2

while if p  a1

  1 ϕ(p)m−4 − (λ − 1)k (−λ − 1)l − (λ − 1)l (−λ − 1)k , 2 √ √ where λ = p if p ≡ 1(mod4) and λ = i p if p ≡ −1(mod4). Under our assumption (1.3), we have m = 0 if p|a1 and m ≤ 1 if p  a1 . By direct computation of the term A(n, p), we can prove that |A(n, p)|  p−1 and hence N (p) = p3 + O(p2 ).  A(n, p) =

Lemma 3.2. (i) For x > 0,  |A(n, q)|  x−1 Aε log60 (x + 2). q>x

So the singular series S(n) := S(n, ∞) is absolutely convergent. (ii) Moreover, we have S(n)  (log log A)−c for some constant c > 0. Proof. Let σ = (log(x + 2))−1 . From Lemmas 6.1 and 6.5 (1) in [4], we have (3.2)



|A(n, q)| ≤

q>x

∞  1−σ  q q=1 −1

x

x 

|A(n, q)| = x−1+σ

∞ 

q 1−σ |A(n, q)|

q=1 1−σ

(1 + p

|A(n, p)|),

p

because xσ  1. Using Lemma 6.6 (1) in [4], we have     60 1−σ (3.3) (1 + p |A(n, p)|) ≤ (1 − p−1−σ )−60 1 + 1+σ ≤ p p pa1 ···a4

pa1 ···a4

= ζ(1 + σ)60  σ −60 = log60 (x + 2). Using (1.3), (3.1) and Lemma 3.1, we get   (3.4) (1 + p1−σ |A(n, p)|) ≤ (1 + cp−σ ) ≤ d(a1 · · · a4 )log2 (1+c)  Aε . p|a1 ···a4

p|a1 ···a4

Now (i) follows from (3.2), (3.3) and (3.4).

SMALL PRIME SOLUTIONS OF A NONLINEAR EQUATION

5

It follows from (1.3), Lemma 3.1 above and Lemma 6.6 (1) in [4] that, for some large constant c > 60, S(n) =

 p







(1 + A(n, p)) 

p|a1 ···a4 p>c

(1 − cp−1 ) 



(1 − cp−1 )



(1 − 60p−2 )

pa1 ···a4 p>c

(1 + p−1 )−(1+c) .

p|a1 ···a4

p|a1 ···a4 p>c

The desired estimate in (ii) now follows from the well-known estimate p−1 )  log log x.



p|x (1

+ 

Proof of Lemma 2.3. We easily derive the following inequalities: 



1≤

a1 m1 +···+a4 m4 =n M <|aj |mj ≤N

1

n−(a1 m1 +···+a3 m3 )≡0( mod |a4 |) M <|aj |mj ≤N,j=1,2,3



=



1

M/|aj |


N N N3 N · · = , |a1 | |a2 | |a3 a4 | |a1 · · · a4 |

where a3 a3 ≡ 1(mod|a4 |). To establish inequalities in the other direction, we first consider case (ii) in which all aj are positive and n = N . If M < aj mj ≤ N/4 for j = 1, 2, 3, then M < N/4 = N − 3(N/4) ≤ N − (a1 m1 + a2 m2 + a3 m3 ) = a4 m4 < N. It follows that 



1≥

a1 m1 +a2 m2 +a3 m3 +a4 m4 =n M
1

n−(a1 m1 +a2 m2 +a3 m3 )≡0( mod a4 ) M
N3 . |a1 · · · a4 |

The case (i) can be treated similarly. We therefore conclude that  a1 m1 +···+a4 m4 =n M <|aj |mj ≤N

1

N3 , |a1 · · · a4 |

from which and the definition of J(n) (in (2.2)) the desired result follows.

4. Estimation of minor arcs In this section, we deal with the estimation of minor arcs. The idea comes from [7]. In m1 , we use the estimates for the linear exponential sum over primes (see (4.6) below). However, the estimates for the quadratic exponential sum over primes

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ZHIXIN LIU

(see (4.12) below) is used in m2 . First, we consider the estimation of  S1 (a1 α)S2 (a2 α)S2 (a3 α)S2 (a4 α)e(−nα)dα m1

 max |S1 (a1 α)|1/2



α∈m1

·



1 0

1 0

4

|S2 (a2 α)| dα

 max |S1 (a1 α)| α∈m1

1/2

|S1 (a1 α)|2 dα

1/4  



1 0

 m1

.

1/4

|S2 (a3 α)|4 dα

1/4  

1 0

1+ε N · , 1/4 |a1 · · · a4 |

|S2 (a4 α)|4 dα

1/4

where we used the following mean-value estimates for S1 (a1 α) and S2 (aj α):  1  N 1+ε  |S1 (a1 α)|2 dα  L2 1 , (4.1) |a1 | 0 m1 =m2 m1 ,m2 ≤N/|a1 |

and for j = 2, 3, 4  1 |S2 (aj α)|4 dα  L4 (4.2) 0



1

2 2 2 m2 1 +m2 =m3 +m4 m2 v ≤N/|aj |,v=1,··· ,4

 N 1+ε . |aj |

Now we turn to the estimation of max |S1 (a1 α)|. The discussion below is similar α∈m1

with which in Lemma 7.1 of [5], and we state it here for completion. Let α ∈ m1 . By Dirichlet’s theorem on Diophantine approximation, there exist integer a, q such that 1≤q≤

N , Q2

(a, q) = 1,

Q2 a . |a1 α − | < q qN

Dividing both sides by |a1 |, the last inequality yields |α −

(4.3)

Q2 a |<  , q qN

where q  is a positive factor of qa1 and (a , q  ) = 1. We claim q  > Q2 . Indeed if q  ≤ Q2 , then by (4.3) and the fact that α ∈ [Q2 /N, 1 + Q2 /N ], we have 1 ≤ a ≤ q  and hence α ∈ M(Q2 ). This contradicts our assumption of α ∈ m1 . Thus N Q2
(4.4)

On the other hand, by the well-known Vinogradov’s Lemma, if y is real and a, q are integers satisfying (a, q) = 1, q ≥ 1 and |y − a/q| < 1/q 2 , then    (log p)e(py)  N q −1/2 + N 4/5 + N 1/2 q 1/2 log4 N. (4.5) S(y) =: M
Hence in view of (4.4) and (4.5),  N N 4/5 N 1/2 1/2 −1/2  4 −1/2 (4.6) S1 (a1 α)  ( )|a1 |1/2 Q2 ) +( ) N Q2 +( log N. |a1 | |a1 | |a1 |

SMALL PRIME SOLUTIONS OF A NONLINEAR EQUATION

Thus, we have



 N 23/56 A5/56

(4.7) m1





1+ε N 2/5   N · |a1 |2/5 |a1 · · · a4 |1/4

+

|a1 |1/4

7

N 3/2 , |a1 ||a2 a3 a4 |1/2 L

provided by N  A15+ε .  Now we come to the estimation of m2 .  S1 (a1 α)S2 (a2 α)S2 (a3 α)S2 (a4 α)e(−nα)dα m2

 max |S2 (a4 α)|



α∈m2

·



1 0

1 0

|S1 (a1 α)|2 dα

|S2 (a2 α)|4 dα

1/4  

1 0



1/2

|S2 (a3 α)|4 dα

1/4

1+ε N  max |S2 (a4 α)| · , α∈m2 |a1 |1/2 |a2 a3 |1/4 where we used (4.1) and (4.2). Now we turn to the estimation of max |S2 (a4 α)|. The discussion below is similar α∈m2

with Lemma 2.2 of [1], and we also state it here for completion. Let α ∈ m2 . By the definition of m2 , there exist integer a, q satisfying (a, q) = 1 such that Q2 , (4.8) Q1 ≤ q ≤ Q2 , |qα − a| < N or Q2 Q1 (4.9) 1 ≤ q ≤ Q2 , < |qα − a| < N N By Dirichlet’s theorem on Diophantine approximation, there exist integers a and q  satisfying (4.10)

1 ≤ q  ≤ (N/a4 )1/2 ,

(a , q  ) = 1,

|q  a4 α − a | < (N/a4 )−1/2 .

Combining (4.8), (4.9) and (4.10), we obtain |q  a4 a − qa | ≤ q  |a4 ||qα − a| + q|q  a4 α − a | < 1, provided N  A15+ε , and hence aa4 a , = q q

and

q =

q . (q, a4 )

Thus by the exponential sums over primes of Ren [6], if y is real and a, q are integers satisfying (a, q) = 1, 1 ≤ a ≤ q, then (4.11)    N 1/2 S(y) =: (log p)e(p2 y)  +N 2/5 +N 1/4 q + N |qy − a| N ε . q + N |qy − a| M
  N  1  = q + N |qα − a| . |a4 | (q, a4 )

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ZHIXIN LIU

In view of (4.11), with α = a4 α, q = q  , and a = a , we have   N N 2/5 N 1/4 (4.12) S2 (a4 α)  ( )1/2 Φ(α)−1/2 + ( ) +( ) Φ(α)1/2 N ε . |a4 | |a4 | |a4 | Recall the definition of m2 and (4.8) and (4.9), we have  N 5/14 1  N 9/40−ε ≤ Φ(α) ≤ . (q, a4 ) A A Thus, we have

 N N N 2/5 N 1/4 N 5/28  ε )1/2 ( )−9/80 |a4 |1/2 + ( ) +( ) ( ) N . S(a4 α)  ( |a4 | A |a4 | |a4 | A

Now we reach    1+ε  N 2/5 N 3/7 N (4.13)  N 31/80 A9/80 + + · 2/5 1/4 5/28 1/2 1/4 |a1 | |a4 | A |a1 | |a3 a4 | m2 

N 3/2 , |a1 ||a2 a3 a4 |1/2 L

provided by N  A15+ε . Then, it therefore follows from (4.7) and (4.13) that

 N 3/2

.



|a1 ||a2 a3 a4 |1/2 L m The contribution from the major arcs can be handled by Lemma 2.1. Now assume the conditions (i) or (ii) in Lemma 2.3. Applying Lemmas 2.2 and 2.3 to Lemma 2.1, we conclude N 3/2 r(n)  |a1 ||a2 a3 a4 |1/2 L 15+ε . This proves Theorems 1.1 and 1.2. provided that N  A 5. Acknowledgement This work is supported by the National Natural Science Foundation of China (Grant No. 11301372), and Specialized Research Fund for the Doctoral Program of Higher Education(Grant No. 20130032120073). References 1. Stephen K.K.Choi and Angel V.Kumchev, Quadratic equations with five prime unknowns, J. Number Theory, 107 (2004), 357-367. 2. L.K. Hua, Some results in the additive prime number theory, Quart. J. Math. (Oxford), 9 (1938), 68-80. 3. T.Y. Li, Enlarged major arcs in the Waring-Goldbach problem, Int. J. Number Theory, 12 (2016), 205-217. 4. W.P. Li, F. Zhao and T.Z. Wang, Small prime solutions of an nonlinear equation, Acta Math. Sinica (Chin. Ser.) 58 (2015), 739-764. (Chinese) 5. M.C. Liu and K.M. Tsang, Small prime solutions of linear equations, in: Theorie des nombres, J.-M. De Koninck and C. Levesque (eds.), de Gruyter, Berlin (1989), 595-624. 6. X.M. Ren, On exponential sums over primes and application in Waring-Goldbach problem, Sci. China Ser. A., 48 (2005), 785-797. 7. L.L. Zhao, The additive problem with one prime and two squares of primes, J. Number Theory, 135 (2014), 8-27.

SMALL PRIME SOLUTIONS OF A NONLINEAR EQUATION

School of mathematics, Tianjin University, Tianjin 300072, P. R. China E-mail address: [email protected]

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