1 Introduction and main result
Additive representations of Waring’s problem over natural numbers by mixtures of squares, cubes and biquadrates are among the more interesting special cases for testing the general expectation that any sufficiently large natural number n is representable in the form $ n=x_1^{k_1}+x_2^{k_2}+\cdots +x_s^{k_s}$ , as soon as $\sum _{j=1}^sk_j^{-1}$ is reasonably large. With the exception of a handful of very special problems, in the current state of knowledge, this sum must exceed $2$ , at the very least, to successfully apply the Hardy–Littlewood method.
In 1999, Brüdern and Wooley [Reference Brüdern and Wooley1] removed a case from the list of those combinations of exponents which have defied treatment. Let $\nu (n)$ denote the number of representations of the natural number n as the sum of a square, four cubes and a biquadrate. Then $\nu (n)\gg n^{{13}/{12}}$ . It is remarkable that this lower bound is of the same order of magnitude as the main term of the conjectured asymptotic formula for $\nu (n)$ predicted by a formal application of the circle method. This result should be compared with the work of Vaughan [Reference Vaughan6], who obtained a theorem of similar strength for the sum of one square and five cubes. The result established by Vaughan [Reference Vaughan6] was strengthened by Cai [Reference Cai2], Li and Zhang [Reference Li and Zhang4] and Xue et al. [Reference Xue, Zhang and Li10].
Based on the result of Brüdern and Wooley [Reference Brüdern and Wooley1], it is reasonable to conjecture that every sufficiently large even integer n can be represented as
where $p_1,\ldots ,p_6$ are prime numbers. This conjecture is probably far outside the reach of current analytic number theory techniques. We shall investigate the exceptional set in the above representation and establish the following result.
Theorem 1.1. Let $E(N)$ denote the number of positive even integers n up to N, which cannot be represented as $n=p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^4$ . Then, for any $\varepsilon>0$ ,
2 Outline of the proof of Theorem 1.1
Let N be a sufficiently large positive integer. By a splitting argument, it is sufficient to consider the even integers $n\in (N/2,N]$ . For the application of the Hardy–Littlewood method, we need to define the Farey dissection. Let $A>0$ be a sufficiently large fixed number, which will be determined at the end of the proof. We set
By Dirichlet’s lemma on rational approximation (see, for example, [Reference Vaughan7, Lemma 2.1]), each $\alpha \in [-1/Q_2,1-1/Q_2]$ can be written in the form
for some integers $a,\,q$ with $1\leqslant a\leqslant q\leqslant Q_2$ and $(a,q)=1$ . Define
Then we obtain the Farey dissection
For $k=2,3,4$ , we define
where $X_k=(N/16)^{{1}/{k}}$ . Let
From (2.2),
To prove Theorem 1.1, we need the following two propositions.
Proposition 2.1. For $n\in (N/2,N]$ ,
where $\mathfrak {S}(n)$ is the singular series defined in (4.1), which is absolutely convergent and satisfies
for any integer n satisfying $n\equiv 0\,(\bmod\ 2)$ and some fixed constant $c^*>0$ ; while $\mathfrak {J}(n)$ is defined by (4.4) and satisfies
The proof of (2.3) in Proposition 2.1 will be demonstrated in Section 4. For the property (2.4) of the singular series, we shall give the proof in Section 5.
Proposition 2.2. Let $\mathcal {Z}(N)$ denote the number of integers $n\in (N/2,N]$ satisfying $n\equiv 0 \pmod 2$ such that
Then
The proof of Proposition 2.2 will be given in Section 6. The rest of this section is devoted to establishing Theorem 1.1 by using Propositions 2.1 and 2.2.
Proof of Theorem 1.1
From Proposition 2.2, we deduce that, with at most $O(N^{{23}/{48}+\varepsilon })$ exceptions, all even integers $n\in (N/2,N]$ satisfy
By Proposition 2.1, we conclude that, with at most $O(N^{{23}/{48}+\varepsilon })$ exceptions, all even integers $n\in (N/2,N]$ can be represented in the form $p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^4$ , where $p_1,p_2,p_3,p_4,p_5,p_6$ are prime numbers. By a splitting argument, we get
This completes the proof of Theorem 1.1.
3 Some auxiliary lemmas
Lemma 3.1 [Reference Ren5, Theorem 1.1]
Suppose that $\alpha $ is a real number and that $|\alpha -a/q|\leqslant q^{-2}$ with $(a,q)=1$ . Let $\beta =\alpha -a/q$ . Then
where $\delta _k={1}/{2}+{\log k}/{\log 2}$ , $d(q)$ is the Dirichlet divisor function and c is a constant.
Lemma 3.2 [Reference Zhao11, Lemma 2.4]
Suppose that $\alpha $ is a real number and that there exist $a\in \mathbb {Z}$ and $q\in \mathbb {N}$ with $(a,q)=1, 1\leqslant q\leqslant Q$ and $|q\alpha -a|\leqslant Q^{-1}$ . If $P^{{1}/{12}}\leqslant Q\leqslant P^{{47}/{12}}$ , then
Lemma 3.3. For $\alpha \in \mathfrak {m}_1$ , we have $f_4(\alpha )\ll N^{{23}/{96}+\varepsilon }$ .
Proof. By Dirichlet’s rational approximation (2.1), for $\alpha \in \mathfrak {m}_1$ , one has $Q_1<q\leqslant Q_2$ and $qX_4^4|\alpha -a/q|\leqslant Q_1$ . From Lemma 3.2,
For $1\leqslant a\leqslant q$ with $(a,q)=1$ , set
For $\alpha \in \mathfrak {m}_2$ , by Lemma 3.1,
say. Then we obtain the following Lemma.
Lemma 3.4. We have
Proof. We have
This completes the proof of Lemma 3.4.
4 Proof of Proposition 2.1
In this section, we establish Proposition 2.1. We first introduce some notation. For a Dirichlet character $\chi \bmod q$ and $k\in \{2,3,4\}$ , we define
where $\chi ^0$ is the principal character modulo q. Let $\chi _2,\chi _3^{(1)},\chi _3^{(2)},\chi _3^{(3)},\chi _3^{(4)},\chi _4$ be Dirichlet characters modulo q. Define
and write
Lemma 4.1 [Reference Vinogradov8, Ch. VI, Problem 14]
For $(a,q)=1$ and any Dirichlet character $\chi~\bmod~q$ ,
with $\beta _k=\log k/\log 2$ , where $d(q)$ is the Dirichlet divisor function.
Lemma 4.2. Let $C_k(q,a)$ be defined as above. Then
Proof. By Lemma 4.1,
Therefore, the left-hand side of (4.2) is
This completes the proof of Lemma 4.2.
To treat the integral on the major arcs, we write $f_k(\alpha )$ as follows:
For the innermost sum on the right-hand side of the above equation, by the Siegel–Walfisz theorem, we have
say. Therefore,
and thus
From this, we derive
Note that
using an elementary estimate. If we extend the interval of the innermost integral over $\lambda $ in (4.3) to $[-1/2,1/2]$ , then the resulting error is
Hence, we obtain
where
Therefore, by (5.4) and Lemma 4.2, (4.3) becomes
which completes the proof of Proposition 2.1.
5 The singular series
In this section, we investigate the properties of the singular series which appears in Proposition 2.1.
Lemma 5.1 [Reference Hua3, Lemma 8.3]
Let p be a prime and $p^\alpha \|k$ . For $(a,p)=1$ , if $\ell \geqslant \gamma (p)$ , then $C_k(p^\ell ,a)=0$ , where
For $k\geqslant 1$ , we define
Lemma 5.2 [Reference Vaughan7, Lemma 4.3]
Suppose that $(p,a)=1$ . Then
where $\mathscr {A}_k$ denotes the set of nonprincipal characters $\chi $ modulo p for which $\chi ^k$ is principal and $\tau (\,\chi )$ denotes the Gauss sum
Also, $|\tau (\,\chi )|=p^{1/2}$ and $|\mathscr {A}_k|=(k,p-1)-1$ .
Lemma 5.3. For $(p,n)=1$ ,
Proof. We denote by $\mathcal {S}$ the sum in the absolute value signs on the left-hand side of (5.1). By Lemma 5.2,
If $|\mathscr {A}_k|=0$ for some $k\in \{2,3,4\}$ , then $\mathcal {S}=0$ . If this is not the case, then
From Lemma 5.2, the sextuple outer sums have not more than
terms. In each of these terms, $ |\tau (\,\chi _2)\tau (\,\chi _3^{(1)})\tau (\,\chi _3^{(2)})\tau (\,\chi _3^{(3)}) \tau (\,\chi _3^{(4)})\tau (\,\chi _4)|=p^3$ . Since in any one of these terms $\overline {\chi _2(a)\chi _3^{(1)}(a)\chi _3^{(2)}(a)\chi _3^{(3)}(a)\chi _3^{(4)}(a)\chi _4(a)}$ is a Dirichlet character $\chi \!\pmod p$ , the inner sum is
Because $\tau (\,\chi ^0)=-1$ for the principal character $\chi ^0\bmod p$ , we have $ |\overline {\chi (-n)}\tau (\,\chi )|\leqslant p^{{1}/{2}}$ . By the above arguments, we obtain
This completes the proof of Lemma 5.3.
Lemma 5.4. Let $\mathcal {L}(p,n)$ denote the number of solutions of the congruence
Then, for $n\equiv 0\!\pmod 2$ , we have $\mathcal {L}(p,n)>0$ .
Proof. We have
where
By Lemma 5.2, $ |E_p|\leqslant (p-1)(\!\sqrt {p}+1)(2\!\sqrt {p}+1)^4(3\!\sqrt {p}+1)$ . It is easy to check that $|E_p|<(p-1)^6$ for $p\geqslant 13$ . Therefore, we obtain $\mathcal {L}(p,n)>0$ for $p\geqslant 13$ . For $p=2,3,5,7,11$ , we can check $\mathcal {L}(p,n)>0$ directly provided that $n\equiv 0 \pmod 2$ .
Lemma 5.5. $A(n,q)$ is multiplicative in q.
Proof. By the definition of $A(n,q)$ in (4.1), we only need to show that $B(n,q)$ is multiplicative in q. Suppose $q=q_1q_2$ with $(q_1,q_2)=1$ . Then
For $(q_1,q_2)=1$ and $k\in \{2,3,4\}$ ,
Putting (5.3) into (5.2), we deduce that
This completes the proof of Lemma 5.5.
Lemma 5.6. Let $A(n,q)$ be as defined in (4.1).
-
(i) We have
(5.4) $$ \begin{align} \sum_{q>Z}|A(n,q)|\ll Z^{-{3}/{2}+\varepsilon}d(n). \end{align} $$ -
(ii) There exists an absolute positive constant $c^*>0$ , such that, for $n\equiv 0\!\pmod 2$ ,
$$ \begin{align*} 0<c^*\leqslant \mathfrak{S}(n)\ll1. \end{align*} $$
Proof. From Lemma 5.5, $B(n,q)$ is multiplicative in q. Therefore,
From (5.5) and Lemma 5.1, $B(n,q)=\prod _{p\|q}B(n,p)$ or $0$ according to if q is square-free or not. Thus,
Write
Then
Applying Lemma 4.1 and noticing that $S_k(p,a)=C_k(p,a)+1$ , we get $S_k(p,a)\ll p^{{1}/{2}}$ , and thus $\mathcal {R}(p,a)\ll p^{{5}/{2}}$ . Therefore, the absolute value of the second term in (5.7) is $\leqslant c_1p^{-{5}/{2}}$ . However, from Lemma 5.3, we can see that the absolute value of the first term in (5.7) is $\leqslant 2^6\cdot 48p^{-{5}/{2}}=3072p^{-{5}/{2}}$ . Let $c_2=c_1+3072$ . Then we have proved that for $p\nmid n$ ,
Moreover, if we use Lemma 4.1 directly, it follows that
and therefore
Let $c_3=\max (c_2,1327104)$ . Then for square-free q,
Hence, by (5.6), we obtain
which proves (5.4), and hence gives the absolute convergence of $\mathfrak {S}(n)$ .
To prove item (ii) of Lemma 5.6, by Lemma 5.5, we first note that
From (5.8),
By (5.9), we obtain
However, it is easy to see that
By Lemma 5.4, $\mathcal {L}(p,n)>0$ for all p with $n\equiv 0\!\pmod 2$ , and thus $1+A(n,p)>0$ . Therefore,
Combining the estimates (5.10)–(5.11), and taking $c^*=c_4c_5c_6>0$ , we derive
This completes the proof Lemma 5.6.
6 Proof of Proposition 2.2
In this section, we shall give the proof of Proposition 2.2. We denote by $\mathcal {Z}_j(N)$ the set of integers n satisfying $n\in (N/2,N]$ and $n\equiv 0 \!\pmod 2$ for which
For convenience, we use $\mathcal {Z}_j$ to denote the cardinality of $\mathcal {Z}_j(N)$ . Also, we define the complex number $\xi _j(n)$ by taking $\xi _j(n)=0$ for $n\not \in \mathcal {Z}_j(N)$ , and
for $n\in \mathcal {Z}_j(N)$ . Plainly, $|\xi _j(n)|=1$ whenever $\xi _j(n)$ is nonzero. Therefore, we obtain
where the exponential sum $\mathcal {K}_j(\alpha )$ is defined by
For $j=1,2$ , set
By [Reference Wooley9, Lemma 2.1] with $k=2$ ,
It follows from Cauchy’s inequality, [Reference Vaughan7, Lemma 2.5], Lemma 3.3 and (6.4) that
Combining (6.3) and (6.5), we get
which implies
Next, we give the upper bound for $\mathcal {Z}_2$ . By (3.2),
say. For $\alpha \in \mathfrak {m}_2$ , either $Q_0^{100}<q\leqslant Q_1$ or $Q_0^{100}<N|q\alpha -a|\leqslant NQ_2^{-1}=Q_1$ . Therefore, by Lemma 3.1,
In view of the fact that $\mathfrak {m}_2\subseteq \mathcal {I}$ , where $\mathcal {I}$ is defined by (3.1), Cauchy’s inequality, the trivial estimate $\mathcal {K}_2(\alpha )\ll \mathcal {Z}_2$ , [Reference Hua3, Theorem 4, page 19], Lemma 3.4 and (6.8), we obtain
where the parameter A is chosen sufficiently large for the bounds (6.8) and (6.9) to work. Moreover, it follows from Cauchy’s inequality, (6.4) and [Reference Hua3, Theorem 4, page 19] that
Combining (6.3), (6.7), (6.9) and (6.10), we deduce that
which implies
From (6.6) and (6.11), $ \mathcal {Z}(N)\ll \mathcal {Z}_1+\mathcal {Z}_2\ll N^{{23}/{48}+\varepsilon }$ . This completes the proof of Proposition 2.2.
Acknowledgement
The authors appreciate the contributions of the referee in reviewing this paper.