Exceptional set in Waring–Goldbach problem for sums of one square and five cubes

1.   Introduction and main result

Waring's problem of mixed powers concerns the representation of sufficiently large integer $n$ in the form

Among the most interesting cases of mixed powers is that of establishing the representations of sufficiently large integer as the sum of one square and $s$ positive cubes for each $s > 1$, i.e.,

In 1930, Stanley ^[7] showed that (1.1) is solvable for $s > 6$. Afterwards, Stanley ^[8] and Watson ^[12] solved the cases $s = 6$ and $s = 5$, respectively. It should be emphasized that Stanley ^[7] obtained the asymptotic formula for $s > 6$, while Sinnadurai ^[6] obtained the asymptotic formula for $s = 6$. But in ^[12], Watson only proved a quite weak lower bound for the number of representation (1.1) with $s = 5$. In 1986, Vaughan ^[9] enhanced Watson's result and derived a lower bound with the expected order of magnitude. In 2002, Wooley ^[13] illustrated that, although the excepted asymptotic formula of (1.1) with $s = 5$ can not be established by the technique currently available, the exceptional set is extremely sparse. To be specific, let $E_1(N)$ denote the number of integer $n\leqslant N$ which can not be represented as one square and five positive cubes with expected asymptotic formula, then Wooley ^[13] showed that $E_1(N)\ll N^\varepsilon$.

In view of the results of Vaughan ^[9] and Wooley ^[13], it is reasonable to conjecture that, for every sufficiently large even integer $N$, the following equation

is solvable. Here and below, the letter $p$, with or without subscript, denotes a prime number. But this conjecture is perhaps out of reach at present. However, it is possible to replace a variable by an almost–prime. In 2014, Cai ^[1] proved that, for every sufficiently large even integer $N$, the following equation

is solvable with $x$ being an almost–prime $\mathcal{P}_{36}$ and the $p_j\, (j = 1, 2, 3, 4, 5)$ primes. Later, in 2018, Li and Zhang ^[3] enhanced the result of Cai ^[1] and showed that (1.3) is solvable with $x$ being an almost–prime $\mathcal{P}_{6}$ and the $p_j\, (j = 1, 2, 3, 4, 5)$ primes. Recently, in 2021, Xue, Zhang and Li ^[15] consider the problem (1.2) with almost equal prime variables, i.e.,

where $U = N^{1-\delta+\varepsilon}$ with $\delta > 0$ hoped to be as large as possible. Let $E(N, U)$ denote the number of all positive even integers $n$ satisfying

which can not be represented as (1.4). One wants to show that there exists $\delta\in(0, 1)$ such that

In ^[15], they proved that $\delta\leqslant8/225$.

In this paper, we shall investigate the exceptional set of the problem (1.2) and establish the following result.

Theorem 1.1. Let $E(N)$ denote the number of positive even integers $n$ up to $N$, which can not be represented as

Then, for any $\varepsilon > 0$, we have

We will establish Theorem 1.1 by using a pruning process into the Hardy–Littlewood circle method. In the treatment of the integrals over minor arcs, we will employ the methods, which is developed by Wooley in ^[14], combining with the new estimates for exponential sum over primes developed by Zhao ^[16]. For the treatment of the integrals on the major arcs, we shall prune the major arcs further and deal with them respectively. The full details will be explained in the following revelent sections.

Notation. Throughout this paper, let $p$, with or without subscripts, always denote a prime number; $\varepsilon$ always denotes a sufficiently small positive constant, which may not be the same at different occurrences. As usual, we use $\varphi(n), d(n)$ and $\omega(n)$ to denote the Euler's function, Dirichlet's divisor function and the number of distinct prime factors of $n$, respectively. Also, we use $\chi\bmod q$ to denote a Dirichlet character modulo $q$, and $\chi^0\bmod q$ the principal character. $e(x) = e^{2\pi i x}$; $f(x)\ll g(x)$ means that $f(x) = O(g(x))$; $f(x)\asymp g(x)$ means that $f(x)\ll g(x)\ll f(x)$. $N$ is a sufficiently large integer and $n\in [N/2, N]$, and thus $\log N\asymp\log n$. The letter $c$, with or without subscripts or superscripts, always denote a positive constant, which may not be the same at different occurrences.

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 (for instance, see Lemma 12 on page 104 of Pan and Pan ^[4]), each $\alpha\in[-1/Q_2, 1-1/Q_2]$ can be written as 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 $k = 2, 3$, we define

where $X_k = (N/16)^{\frac{1}{k}}$. Let

From (2.2), one has

In order to prove Theroem 1.1, we need the two following propositions:

Proposition 2.1. For $n\in[N/2, N]$, there holds

where $\mathfrak{S}(n)$ is the singular series defined in (4.1), which is absolutely convergent and satisfies

for any integer $n$ satisfying $n\equiv 0\!\pmod 2$ and some fixed constant $c^* > 0$; while $\mathfrak{J}(n)$ is defined by (4.5) and satisfies

The proof of (2.3) in Proposition 2.1 will be demonstrated in Section 4. For the property (2.4) of 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 $n\equiv0\!\pmod 2$ such that

Then we have

The proof of Proposition 2.2 will be given in section 6. The remaining part of this section is devoted to establishing Theorem 1.1 by using Proposition 2.1 and Proposition 2.2.

Proof of Theorem 1.1. From Proposition 2.2, we deduce that, with at most $O\big(N^{\frac{4}{9}+\varepsilon}\big)$ exceptions, all even integers $n\in[N/2, N]$ satisfy

from which and Proposition 2.1, we conclude that, with at most $O\big(N^{\frac{4}{9}+\varepsilon}\big)$ exceptions, all even integers $n\in[N/2, N]$ can be represented in the form $p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3$, 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. 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 we have

where $\delta_k = \frac{1}{2}+\frac{\log k}{\log2}$ and $c$ is a constant.

Proof. See Theorem 1.1 of Ren ^[5].

Lemma 3.2. Suppose that $\alpha$ is a real number, and that there exist $a\in\mathbb{Z}$ and $q\in\mathbb{N}$ with

If $P^{\frac{1}{2}}\leqslant Q\leqslant P^{\frac{5}{2}}$, then one has

Proof. See Lemma 8.5 of Zhao ^[16].

Lemma 3.3. For $\alpha\in\mathfrak{m}_1$, we have

Proof. By Dirichlet's rational approximation (2.1), for $\alpha\in\mathfrak{m}_1$, one has $Q_1\leqslant q\leqslant Q_2$. From Lemma 3.2, we get

This completes the proof of Lemma 3.3.

For $1\leqslant a\leqslant q$ with $(a, q) = 1$, set

For $\alpha\in\mathfrak{m}_2$, by Lemma 3.1, we have

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 shall concentrate on establishing Proposition 2.1. We first introduce some notations. For a Dirichlet character $\chi\bmod q$ and $k = 2, 3$, we define

where $\chi^0$ is the principal character modulo $q$. Let $\chi_2, \chi_3^{(1)}, \chi_3^{(2)}, \chi_3^{(3)}, \chi_3^{(4)}, \chi_3^{(5)}$ be Dirichlet characters modulo $q$. Define

and write

Lemma 4.1. For $(a, q) = 1$ and any Dirichlet character $\chi\bmod q$, there holds

with $\beta_k = (\log k)/\log2$.

Proof. See the Problem 14 of Chapter VI of Vinogradov ^[11].

Lemma 4.2. Let $C_k(q, a)$ be defined as above. Then there holds

Proof. By Lemma 4.1, we have

Therefore, the left–hand side of (4.2) is

This completes the proof of Lemma 4.2.

In order 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 Siegel–Walfisz theorem, we have

say. Therefore, we have

and thus

Then we derive that

Noting that

hence the innermost integral in (4.3) can be written as

By using the elementary estimate

we know that if we extend the interval of the integral in (4.4) to $[-1/2, 1/2]$, then the resulting error is

Hence we obtain

where

Therefore, by (5.4), (4.3) becomes

which completes the proof of Proposition 2.1.

5.   The singular series

In this section, we shall concentrate on investigating the properties of the singular series which appear in Proposition 2.1.

Lemma 5.1. Let $p$ be a prime and $p^\alpha\|k$. For $(a, p) = 1$, if $\ell\geqslant\gamma(p)$, we have $C_k(p^\ell, a) = 0$, where

Proof. See Lemma 8.3 of Hua ^[2].

For $k\geqslant1$, we define

Lemma 5.2. Suppose that $(p, a) = 1$. Then

where $\mathscr{A}_k$ denotes the set of non–principal characters $\chi$ modulo $p$ for which $\chi^k$ is principal, and $\tau(\chi)$ denotes the Gauss sum

Also, there hold $|\tau(\chi)| = p^{1/2}$ and $|\mathscr{A}_k| = (k, p-1)-1$.

Proof. See Lemma 4.3 of Vaughan ^[10].

Lemma 5.3. For $(p, n) = 1$, we have

Proof. We denote by $\mathcal{S}$ the left–hand side of (5.1). By Lemma 5.2, we have

If $|\mathscr{A}_k| = 0$ for some $k\in\{2, 3\}$, then $\mathcal{S} = 0$. If this is not the case, then

From Lemma 5.2, the sextuple outer sums have not more than $\big((2, p-1)-1\big)\times\big((3, p-1)-1\big)^5 \leqslant1\times2^5 = 32$ terms. In each of these terms, we have

Since in any one of these terms

is a Dirichlet character $\chi\!\pmod p$, the inner sum is

From the fact that $\tau(\chi^0) = -1$ for principal character $\chi^0\bmod p$, we have

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, we obtain

It is easy to check that $|E_p| < (p-1)^6$ for $p\geqslant13$. 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.

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 we have

For $(q_1, q_2) = 1$ and $k\in\{2, 3\}$, there holds

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). Then

$\rm{(i)}$ We have

$\rm{(ii)}$ There exists an absolute positive constant $c^* > 0$, such that, for $n\equiv0\pmod2$, there holds

Proof. From Lemma 5.5, we know that $B(n, q)$ is multiplicative in $q$. Therefore, there holds

From (5.5) and Lemma 5.1, we deduce that $B(n, q) = \prod\limits_{p\|q}B(n, p)$ or $0$ according to whether $q$ is square–free or not. Thus, one has

Write

Then

Applying Lemma 4.1 and noticing that $S_k(p, a) = C_k(p, a)+1$, we get $S_k(p, a)\ll p^{\frac{1}{2}}$, and thus $\mathcal{R}(p, a)\ll p^{\frac{5}{2}}$. Therefore, the second term in (5.7) is $\leqslant c_1p^{-\frac{5}{2}}$. On the other hand, from Lemma 5.3, we can see that the first term in (5.7) is $\leqslant2^6\cdot 32p^{-\frac{5}{2}} = 2048p^{-\frac{5}{2}}$. Let $c_2 = c_1+2048$. Then we have proved that, for $p\nmid n$, there holds

Moreover, if we use Lemma 4.1 directly, it follows that

and therefore

Let $c_3 = \max(c_2, 995328)$. Then for square–free $q$, we have

Hence, by (5.6), we obtain

which proves (5.4), and hence gives the absolutely convergence of $\mathfrak{S}(n)$. In order to prove (ii) of Lemma 5.6, by Lemma 5.5, we first note that

From (5.8), we have

By (5.9), we obtain

On the other hand, it is easy to see that

By Lemma 5.4, we know that $\mathcal{L}(p, n) > 0$ for all $p$ with $n\equiv0\!\pmod 2$, and thus $1+A(n, p) > 0$. Therefore, there holds

Combining the estimates (5.10)–(5.13), and taking $c^* = c_4c_5c_6 > 0$, we derive that

Moreover, by (5.8) and (5.9), we have

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 $n\equiv0 \!\pmod 2$ for which the following estimate

holds. For convenience, we use $\mathcal{Z}_j$ to denote the cardinality of $\mathcal{Z}_j(N)$ for abbreviation. Also, we define the complex number $\xi_j(n)$ by taking $\xi_j(n) = 0$ for $n\not\in\mathcal{Z}_j(N)$, and when $n\in\mathcal{Z}_j(N)$ by means of the equation

Plainly, one has $|\xi_j(n)| = 1$ 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

From (6.1)–(6.3), we derive that

By Lemma 2.1 of Wooley ^[14] with $k = 2$, we know that, for $j = 1, 2$, there holds

It follows from Cauchy's inequality, Lemma 2.5 of Vaughan ^[10], Lemma 3.3 and (6.5) that

Combining (6.4) and (6.6), we get

which implies

Next, we give the upper bound for $\mathcal{Z}_2$. By (3.2), we obtain

say. For $\alpha\in\mathfrak{m}_2$, then either one has $Q_0^{100} < q\leqslant Q_1$ or $Q_0^{100} < N|q\alpha-a|\leqslant NQ_2^{-1} = Q_1$. Therefore, by Lemma 3.1, we get

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 $\mathcal{K}_2(\alpha)\ll\mathcal{Z}_2$, Theorem 4 of Hua (See ^[2], P 19), Lemma 3.4 and (6.9), we obtain

where the parameter $A$ is chosen sufficiently large for above bound to work. Moreover, it follows from Cauchy's inequality, (6.5) and Theorem 4 of Hua (See ^[2], P 19) that

Combining (6.4), (6.8), (6.10) and (6.11), we deduce that

which implies

From (6.7) and (6.12), we have

This completes the proof of Proposition 2.2.

Acknowledgments

The authors would like to express the most sincere gratitude to the referee for his/her patience in refereeing this paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11901566, 12001047, 11971476, 12071238), the Fundamental Research Funds for the Central Universities (Grant No. 2021YQLX02), the National Training Program of Innovation and Entrepreneurship for Undergraduates (Grant No. 202107010), the Undergraduate Education and Teaching Reform and Research Project for China University of Mining and Technology (Beijing) (Grant No. J210703), and the Scientific Research Funds of Beijing Information Science and Technology University (Grant No. 2025035).

Conclict of interest

The authors declare that no conflict of interest exists in this manuscript.