On the Waring-Goldbach problem for two squares and four cubes

: Let N be a su ﬃ ciently large integer. In this article, it is proved that, with at most + O N ε 112 ( ) exceptions, all even positive integers up to N can be represented in the form


(
) exceptions, all even positive integers up to N can be represented in the form

Introduction and main result
In 1938, Hua [1] proved that every sufficiently large integer N satisfying ≡ N 5 mod 24 ( )can be represented as the sum of five squares of primes, while every sufficiently large integer N satisfying ≡ N 1 mod 2 ( )can be represented as the sum of nine cubes of primes.Based on the significant results of Hua, it seems reasonable to conjecture that every sufficiently large integer satisfying some necessary congruence conditions can be written as the sum of four squares of primes or eight cubes of primes, i.e.On the other hand, Hooley [2] introduced divisor sum techniques into the investigation of Waring's problem of mixed powers.In particular, Hooley's techniques provide an asymptotic formula for every sufficiently large integer N as the sum of two squares and four cubes of positive integers.Moreover, motivated by Hooley's result, it is reasonable to conjecture that every sufficiently large even integer N can be represented as follows: where p p ,…, 1 6 are primes.Meanwhile, we can regard equation (1.3) as the hybrid conjecture of equations (1.1) and (1.2).But this expectation is probably far out of the reach of modern number theory techniques.
In 2016, Cai [3] gave an approximation to the conjecture (1.3) and proved that any sufficiently large even integer N can be written in the form almost-prime 3 .Afterwards, Cai's studies [3] in this direction were subsequently generalized by Zhang and Li [4].On the other hand, in 2019, Liu [5] considered the exceptional set of the conjecture (1.3) and showed that ( ) where E N ( ) denotes the number of positive even integers n up to N , which cannot be repre- sented as In this article, we shall continue to consider the exceptional set of the problem (1.3) and improve the previous result.Then, for any > ε 0, we have 2 Preliminary and outline of the proof of Theorem 1.1 In order to better illustrate Lemmas 2.1 and 2.2, we first introduce some notations and definitions.
When ⊆ , we write for the complement ⧹ of within .When a and b are non-negative integers, it is convenient to denote by It is convenient, when k is a natural number, to describe a subset of as being a high-density subset of the kth powers when (i) one has ⊆ ∈ n n : k { } , and (ii) for each positive number ε, whenever N is a natural number sufficiently large in terms of ε, then | | .In addition, when > θ 0, we shall refer to a set ⊆ as having complementary density growth exponent smaller than θ when there exists a positive number δ with the property that, for all sufficiently large natural numbers N , one has . When q is a natural number and ∈ − q 0, 1, …, 1 a { } , we define = q , a a by = + ∈ mq m : .q , a a { } In addition, we describe a set as being a union of arithmetic progressions modulo q when, for some subset L of − q 0, 1, …, 1 { } , one has In such circumstances, given a subset of and integers a and b, it is convenient to write Let be a union of arithmetic progressions modulo q, for some natural number q.When k is a natural number, we describe a subset of as being a high-density subset of the kth powers relative to when (i) one has ⊆ ∈ n n : k { } , and (ii) for each positive number ε, whenever N is a natural number sufficiently large in terms of ε, then In addition, when > θ 0, we shall refer to a set ⊆ as having -com- plementary density growth exponent smaller than θ when there exists a positive number δ with the property that, for all sufficiently large natural numbers N , one has Lemma 2.1.Let , , and be unions of arithmetic progressions modulo q, for some natural number q, and suppose that ⊆ + . Suppose also that is a high-density subset of the squares relative to , and that ⊆ has -complementary density growth exponent smaller than 1.Then, whenever > ε 0 and N is a natural number sufficiently large in terms of ε, one has Proof.See Theorem 2.2 of Kawada and Wooley [6].□ Lemma 2.2.Let , , and be unions of arithmetic progressions modulo q, for some natural number q, and suppose that ⊆ + .Suppose also that is a high-density subset of the cubes relative to , and that ⊆ has -complementary density growth exponent smaller than θ, for some positive number θ.Then, whenever > ε 0 and N is a natural number sufficiently large in terms of ε, without any condition on θ, one has Proof.See Theorem 4.1 (a) of Kawada and Wooley [6].□ In order to prove Theorem 1.1, we need the following proposition, whose proof will be given in Section 3. 3 .Then, for any > ε 0, we have The remaining part of this section is devoted to establishing Theorem 1.  3 Thus, we have Then, is an arithmetic progression modulo 2, and so are and .In addition, there hold ⊆ + and ⊆ + .Moreover, it follows from the prime number theorem in arithmetic progression that Therefore, 2 is a high-density subset of the squares relative to , while 3 is a high-density subset of the cubes relative to .By Proposition 2.3, it is easy to see that and thus 1 has -complementary density growth exponent smaller than 1.From Lemma 2.1, we know that Let the integers N j for ⩾ j 0 be determined by the iterative formula where ⌈ ⌉ N denotes the least integer not smaller than N .Moreover, we define J to be the least positive integer with the property that ⩽ N 10 j , then ≪ J N log .Therefore, there holds On the Waring-Goldbach problem for two squares and four cubes  3 By equation (2.2), we know that and thus 2 has -complementary density growth exponent smaller than 1 2 .From Lemma 2.2, we obtain .
Therefore, with the same notation of equation (2.1), we deduce that which completes the proof of Theorem 1.1.□ 3 Outline of the proof of Proposition 2.3 In this section, we shall give an outline of the proof of Proposition 2.3.Let N be a sufficiently large positive integer.For = k 2, 3, we define

R( ) ( )( )( )( )
Then, for any > Q 0, it follows from the orthogonality that In order to apply the circle method, we set By Dirichlet's lemma on rational approximation (e.g.see Lemma 2.1 of Vaughan [7]), each for some integers a q , with ⩽ ⩽ ⩽ a q Q 1 , and ) .Then, we define the major arcs M and minor arcs m as follows: q P a q a q where Then, one has In order to prove Proposition 2.3, we need the following two propositions, whose proofs will be given in Sections 4 and 6, respectively.
where n S( ) is the singular series defined in equation (4.1), which is absolutely convergent and satisfies for any integer n satisfying ≡ n 0 mod 2 ( ) and some fixed constant > * c 0, while n J( ) is defined by equation (4.9) and satisfies

J( )
For the properties (3.3) of singular series, we shall give the proof in Section 5.
Proposition 3.2.Let the minor arcs m be defined as in equation (3.2) with P and Q defined in equation (3.1).Then, we have The remaining part of this section is devoted to establishing Proposition 2.3 by using Propositions 3.1 and 3.2.
By Bessel's inequality, we have On the Waring-Goldbach problem for two squares and four cubes  5 Combining equations (3.4) and (3.5) and Proposition 3.2, we have Therefore, with at most from which, using Proposition 3.1, we deduce that, with at most ) can be represented in the form 3 , where p p p , , 1 2 3 , and p 4 are prime numbers.By a splitting argument, we obtain This completes the proof of Proposition 2.3.□ 4 Proof of Proposition 3.1 In this section, we shall concentrate on proving Proposition 3.1.We first introduce some notations.For a Dirichlet character χ q mod and ∈ k 2, 3 { }, we define ( ) , and χ 3 3 and write ) and any Dirichlet character χ q mod , there holds Proof.See Problem 14 of Chapter VI of Vinogradov [8].□ Lemma 4.2.The singular series n S( ) satisfies equation (3.3).
The proof of Lemma 4.2 is provided in Section 5.
[ ] ( ) ( ) ( ) , and χ 0 be the principal character modulo q.Then, there holds Proof.By Lemma 4.1, we have a a q q 2 0 3 Therefore, the left-hand side of equation (4.2) is This completes the proof of Lemma 4.4.
where = δ 1 χ or 0 according to whether χ is principal or not.Then, by the orthogonality of Dirichlet characters, for = a q , 1 ( ) , we have , we define the sets j S as follows: In addition, we write = ⧹ 2, 3, 3, 3 } .Then, we have where In the following content of this section, we shall prove that I 1 produces the main term, while the others contribute to the error term.
For = k 2, 3, applying Lemma 4.3 to V λ k ( ), we have On the Waring-Goldbach problem for two squares and four cubes  7 Putting equation (4.5) into I 1 , we see that By using the elementary estimate and Lemma 4.4 with = r 1 0 , the O-term in equation (4.6) can be estimated as follows:

| ( )| ( )
If the interval of the integral in the main term of equation ( 4 then from equation (3.1), we can see that the resulting error is where In order to estimate the contribution of I j for = j 2, 3,…, 8, we shall need the following three preliminary lemmas, i.e.Lemmas 4.5-4.7,whose proofs are exactly the same as Lemmas 3.5-3.7 in Zhang and Li [10], so we omit the details herein.In view of this, for ∈ k 2, 3 { }, we recall the definition of W χ λ , k ( ) in equation (4.3) and write Here and below, * Σ indicates that the summation is taken over all primitive characters.
Lemma 4.5.Let P and Q be defined as in equation (3.1).Then, we have ( ) Lemma 4.6.Let P and Q be defined as in equation (3.1).Then, we have ( ) Lemma 4.7.Let P and Q be defined as in equation (3.1).Then, for any > A 0, we have Now, we concentrate on estimating the terms I j for = j 2, 3,…, 8.We begin with the term I 8 , which is the most complicated one.Reducing the Dirichlet characters in I 8 into primitive characters, we have q P a a q q qQ qQ χ q χ q q P χ q χ qχ qχ q where χ 0 is the principal character modulo q and = r r r r r , , , ) .From this and the definition of for primitive characters χ 2 and χ i 3 On the Waring-Goldbach problem for two squares and four cubes  9 In the last integral, we pick out W χ λ , and then use Cauchy's inequality to derive that .
Now we introduce the iterative procedure to bound the sums over r r r , , the sum over r 3 By Lemma 4.5 again, the contribution of the quantity on the right-hand side of equation (4.11) to the sum over r 3 By Lemma 4.6, the contribution of the quantity on the right-hand side of equation (4.12) to the sum over Finally, from Lemma 4.7, inserting the bound on the right-hand side of equation (4.13) to the sum over r 2 in equation (4.10), we obtain For the estimation of the terms Using this estimate and the upper bound of V λ k ( ), which derives from equations (4.5) and (4.7), that ≪ V λ N

The singular series
In this section, we shall investigate the properties of the singular series that appear in Proposition 3.1.
Lemma 5.1.Let p be a prime and p k , where Proof.See Lemma 8.3 of Hua [11].□ For ⩾ k 1, we define where k A denotes the set of non-principal characters χ modulo p for which χ k is principal, and τ χ ( ) denotes the Gauss sum In addition, there hold Proof.See Lemma 4.3 of Vaughan [7].On the Waring-Goldbach problem for two squares and four cubes  11 Proof.We denote by Σ the left-hand side of equation (5.1).By Lemma 5.2, we have From Lemma 5.2, the quadruple outer sums have not more than eight terms.In each of these terms, we have Since in any one of these terms χ a χ a χ a χ a ), the inner sum is χ a e an p χ n χ an e an p χ n τ χ .

| ( ) ( )|
By the above arguments, we obtain ) denote the number of solutions to the following congruence: ( ) Proof.We have where ) is multiplicative in q.
Proof.By the definition of A n q , ( ) in equation ( 4.1), we only need to show that B n q , ( ) is multiplicative in q.Suppose that = q q q 1 2 with = q q , 1 1 2

( )
. Then, we have B n q q C q q a C q q a e an q q C q q a q a q C q q a q a q e a n q e a n q , , , , , .
a a q q q q a a q q a a q q 1 2 1 , 1 C q q a q a q e a q a q m q q e a q a q m q m q q q e a m q q e a m q q C q a C q a , , , .
Putting equation ( 5.3) into equation (5.2), we deduce that B n q q C q a C q a e a n q C q a C q a e a n q This completes the proof of Lemma □ Lemma 5.6.Let A n q , ( ) be defined as in equation (4.1).Then, (i) we have and thus the singular series n S( ) is absolutely convergent and satisfies for any integer n satisfying ≡ n 0 mod 2 ( ).
Proof.From Lemma 5.5, we know that B n q , ( ) is multiplicative in q.Therefore, there holds B n q B n p C p a C p a e an p , , , , .
From equation (5.4) and Lemma 5.1, we deduce that ‖ or 0 according to whether q is square-free or not.Thus, we have On the Waring-Goldbach problem for two squares and four cubes  13

V(
) .Therefore, the second term in equation ( 5 .Then, we have proved that, for ∤ p n, there holds Moreover, if we use Lemma 4.1 directly, it follows that and therefore, ).Then, for square-free q, we have p q p n p q p n p q p n p q p n ω q p q p n q ε Hence, by equation (5.5), we obtain This proves (i) of Lemma 5.6.
To prove (ii) of Lemma 5.6, by Lemma 5.5, we first note that | (5.9) From equation (5.7), we have  In this section, we first present some lemmas that will be used to prove Proposition 3.2.
Proof.See Theorem 1 of Ren [13].□ Lemma 6.3.Suppose that α is a real number, and that there exist ∈ a and ∈ q with On the Waring-Goldbach problem for two squares and four cubes  15 5 2 , then we have Proof.See Lemma 8.5 of Zhao [14].□ Lemma 6.4.Suppose that α is a real number, and that there exist integers ∈ a and ∈ q satisfying 5 2 , then we have , by Dirichlet's lemma on rational approximation (for instance, see Lemma 2.1 of Vaughan [7]), there exist integers ∈ a and ∈ q such that ) .Next, we shall discuss the upper bound of f α 3 ( ) according to the size of q and β | |.
| | , we have Then, by Lemma 6.2, we obtain which is acceptable.
Case 2 If > ∕ q X 3 1 2 , it follows from Lemma 6.3 that | | , which combined with Lemma 6.3 yields that Combining the above three cases, we derive the desired conclusion of Lemma 6.4.□ Lemma 6.5.Let f α k ( ) be defined as above.Then, we have with By Lemma 2.5 of Vaughan [7], we have and thus the contribution with = x x . Combining the above two cases, we deduce that This completes the proof of Lemma 6.5.□ Define the multiplicative function w q 3 ( ) by One has uniformly for ∈ γ that A ( ] , we define Lemma 6.7.Let M be the union of the intervals q a , M( ) , where = − ⩽ − ∕ q a α qα a X , : .

M( ) { | | }
Suppose that G α ( ) and h α ( ) are integrable functions of period one.Let = g α g α A ( ) ( ) be given in equation (6.1), and let ⊆ 0, 1 m [ ) be a measurable set.Then, we have where On the Waring-Goldbach problem for two squares and four cubes  17 Proof.See Lemma 3.1 of Zhao [14].

□
For the proof of Proposition 3.2, we define a general Hardy-Littlewood dissection employed in our application of the circle method.When X is a positive number with ⩽ X N , we take X N( ) to be the union of the intervals ) .In addition, when ⩽ Finally, we take . Therefore, by Lemmas 6.1 and 6.4, it is easy to obtain where  ( ) (6.9) We define the function , and when ∈ α N N , by writing = + − ∕ − α q qN α a q Ξ .

Theorem 1 . 1 .
Let E N ( ) denote the number of positive even integers n up to N , which cannot be represented as ϖ 0. Therefore, by Lemma 4.2, equation (4.6) becomes

1 3 2 .
.6) is ⩽ − c p On the other hand, from Lemma 5.3, we can see that the first term in equation (5.6) is ⩽ Combining the estimates (5.9)-(5.12)and taking = > *

16
Proof.Trivially, the conclusion can be deduced by counting the number of solutions to the underlying Diophantine equation Min Zhang et al.

∈ α 3 m
, by Lemmas 6.1 and 6.4, we obtain On the Waring-Goldbach problem for two squares and four cubes  19 .Moreover, Lemma 2 of Brü dern[15] supplies the following upper bound: argument, from equations (6.12) and (6.13), we derive that 1 by using Lemmas 2.1 and 2.2 and Proposition 2.3.