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Slim exceptional sets of Waring-Goldbach problems involving squares and cubes of primes X. Han, H. Liu School of Mathematics and Statistics, Shandong Normal University, Jinan, P.R. China
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     § 1. IntroductionThe famous Waring problem with mixed powers is concerned with the solvability of the equation in natural numbers $n_1, \dots, n_s$ for a sufficiently large integer $n$, where $k_1, \dots, k_s$ are integers such that $k_s\geqslant \dots\geqslant k_2 \geqslant k_1\geqslant 2$. Results of this kind are not well known. We refer to [1]–[4] for details.In 1969 Vaughan [5] proved that any sufficiently large integer $n$ can be represented as the sum of two squares and four cubes of natural numbers by establishing that
$$
\begin{equation}
\Xi(n)=\frac{\Gamma^2(3/2)\Gamma^4(4/3) }{\Gamma(7/3)} \mathfrak{S}^*(n)n^{4/3}+ O(n^{4/3-1/48+\varepsilon}),
\end{equation}
\tag{1.2}
$$
where $\Xi(n)$ represents the number of the representations of $n$ in the form and
$$
\begin{equation*}
\mathfrak{S}^*(n)=\sum_{q=1}^{\infty}\frac{1}{q^{6}}\sum_{\substack{a=1 \\ (a, q)=1}}^{q} \biggl(\sum_{r_{1}=1}^{q}e\biggl(\frac{ar_{1}^{2} }{q}\biggr)\biggr)^{2} \biggl(\sum_{r_{2}=1}^{q}e\biggl(\frac{ar_{2}^{3} }{q}\biggr)\biggr)^{4}e \biggl(-\frac{an}{q}\biggr).
\end{equation*}
\notag
$$
Motivated by the works of Vinogradov [6] and Hua [7], many scholars began to study the Waring-Goldbach problem by restricting the choice of natural numbers $n_1, \dots, n_s$ to primes $p_1, \dots, p_s$ and putting some local conditions on $n$. Then in view of Vaughan’s result it is reasonable to conjecture that every sufficiently large even integer $n$ can be represented as the sum of two squares and four cubes of primes, that is, It seems that this conjecture is out of reach now. But some approximations have been studied in this direction. Denote by $P_r$ an almost prime with at most $r$ prime factors, counted in accordance with multiplicities. Motivated by the works of Brüdern [8], [9], combining sieve methods and the Hardy-Littlewood method, Cai [10] proved that all sufficiently large even integers $n$ can be represented in the form with $x$ being a $P_3$.Set
$$
\begin{equation}
\begin{aligned} \, \mathscr{E}=\bigl\{n\in\mathbb{N}\colon 2\mid n, \, n\neq p_{1}^{2}+p_{2}^{2}+p_{3}^{3} +p_{4}^{3}+p_{5}^{3}+p_{6}^{3}\bigr\}. \end{aligned}
\end{equation}
\tag{1.4}
$$
Let $E(N)$ denote the cardinality of $\mathscr{E}\cap(0,N]$. Recently, Y. Liu [11] considered the set of possible exceptions to the representation (1.3) and proved that, for any ${\varepsilon>0}$, In this paper, we improve further the above result of Y. Liu by establishing the following theorem. In comparison, $1/4=0.25$ and $1/12=0.0833\dots$ . It is easy to find that Theorem 1 improves on Y. Liu’s result by a factor of $3$. It is also interesting to consider a variant problem by replacing one square of a prime in (1.3) by the cube of a prime, that is,
$$
\begin{equation*}
n=p_{1}^{2}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}+p_{5}^{3}+p_{6}^{3}.
\end{equation*}
\notag
$$
In 2014, Cai [12] proved that any sufficiently large even integer $n$ can be represented in the form with $x$ being a $P_{36}$. Subsequently, the almost prime $P_{36}$ was improved to $P_{6}$ by Li and Zhang [13]. Set
$$
\begin{equation}
\mathscr{E}_1=\bigl\{n\in\mathbb{N}\colon 2\mid n,\, n\neq p_{1}^{2}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}+p_{5}^{3}+p_{6}^{3}\bigr\}.
\end{equation}
\tag{1.5}
$$
Let $E_1(N)$ denote the cardinality of $\mathscr{E}_1\cap(0,N]$. Applying an argument analogous to the proof of Theorem 1 we can obtain the following theorem. We establish Theorems 1 and 2 by applying the Hardy-Littlewood method. Our improvement benefits from the works of Kawada and Wooley [14] (see Lemmas 1–3 below) and Zhao [15] (see Lemma 9 in what follows). Unlike the treatment in [11], first we consider a Waring-Goldbach problem related to Theorems 1 and 2, but with fewer summands. We establish a result on an exceptional set for a related Waring-Goldbach problem with fewer summands. Then, with the help of arguments in [14] and [15], we obtain a better result. Due to the fact that the proof of Theorem 2 is similar to that of Theorem 1, we just sketch its proof in § 5. In § 2 we give the proof of Theorem 1. In §§ 3 and 4, we present several lemmas and the proof of Proposition 2, respectively. Notation. Throughout this paper, the letter $p$, with or without a subscript, always represents a prime number; $\varepsilon$ denotes a sufficiently small positive constant, which is not necessarily the same at each occurrence. We use $\chi \pmod q$ to denote a Dirichlet character modulo $q$ and $\chi^{0} \pmod q$ to denote the principal character; $f(x)\ll g(x)$ and $f(x)\asymp g(x)$ mean that $f(x)=O(g(x))$ and $f(x)\ll g(x)\ll f(x)$, respectively; $d(n)$ is the divisor function. The letter $c$, with or without subscripts or superscripts, always denotes a positive constant. As usual, we abbreviate $e^{2\pi ix}$ and $\log N$ to $e(x)$ and $L$, respectively. § 2. Preliminaries and outline of the methodTo explain Lemmas 1–3 we need to introduce further notation. Let $N$ be a sufficiently large natural number and $\mathbf{A}$ be a subset of $\mathbb{N}$. We take $\overline{\mathbf{A}}$ to be the complement $\mathbb{N}\setminus \mathbf{A}$ of $\mathbf{A}$ to $\mathbb{N}$. For a general interval $(a, b]$ we denote by $(\mathbf{A})_{a}^{b}$ the set $\mathbf{A}\cap(a, b]$ and by $|\mathbf{A}|_{a}^{b}$ the cardinality of $\mathbf{A}\cap(a,b]$. Let $|\overline{\mathbf{A}}|_{a}^{b}$ be the number of natural numbers in the interval $(a,b]$ that do not belong to $\mathbf{A}$. For $\mathbf{A},\mathbf{B}\subseteq\mathbb{N}$ we write
$$
\begin{equation*}
\mathbf{A}\pm\mathbf{B}=\bigl\{a\pm b\colon a\in\mathbf{A},\ b\in\mathbf{B}\bigr\}.
\end{equation*}
\notag
$$
When $k\in\mathbb{N}$, a subset $\mathbf{Q}$ of $\mathbb{N}$ is a high-density subset of the $k$th powers if (i) $\mathbf{Q}\subseteq\{n^{k},n\in\mathbb{N}\}$, (ii) $|\mathbf{Q}|_{0}^{N}>N^{1/k-\varepsilon}$. For $\theta>0$ a set $\mathbf{R}\subseteq\mathbb{N}$ is said to have the complementary density growth exponent smaller than $\theta$ if there exists a positive number $\delta$ such that $|\overline{\mathbf{R}}|<N^{\theta-\delta}$. When $q\in\mathbb{N}$ and $\mathbf{a}\in \{0,1,\dots,q-1\}$, let $\mathcal{P}_{\mathbf{a}}=\mathcal{P}_{\mathbf{a},q}$ denote
$$
\begin{equation*}
\mathcal{P}_{\mathbf{a},q}=\{\mathbf{a}+mq\colon m\in\mathbb{Z}\}.
\end{equation*}
\notag
$$
We call a set $\mathbf{L}$ a union of arithmetic progressions modulo $q$ if
$$
\begin{equation*}
\mathbf{L}=\bigcup_{\mathbf{l}\in\mathfrak{L}}P_{\mathbf{l},q}
\end{equation*}
\notag
$$
for some subset $\mathfrak{L}$ of $\{0,1,\dots,q-1\}$. Also let
$$
\begin{equation*}
\langle\mathbf{C}\wedge\mathbf{L}\rangle_{a}^{b} =\min_{\mathbf{l}\in\mathfrak{L}}|\mathbf{C}\cap P_{\mathbf{l},q}|_{a}^{b},
\end{equation*}
\notag
$$
where $\mathbf{C}\subseteq\mathbb{N}$ and $a,b\in\mathbb{Z}$. When $k\in\mathbb{N}$ and $\mathbf{L}$ is a union of arithmetic progressions modulo $q$, a subset $\mathbf{Q}$ of $\mathbb{N}$ is a high-density subset of the $k$th powers relative to $\mathbf{L}$ if (i) $\mathbf{Q}\subseteq\{n^{k},n\in\mathbb{N}\}$, (ii) $\langle \mathbf{Q} \wedge \mathbf{L} \rangle_{0}^{N} \gg_{q}N^{1/k-\varepsilon}$. For $\theta>0$ a set $\mathbf{R}\subseteq\mathbb{N}$ is said to have an $\mathbf{L}$-complementary density growth exponent smaller than $\theta$ if $|\overline{\mathbf{R}}\cap\mathbf{L}|_{0}^{N}<N^{\theta-\delta}$. Lemma 1 (see [14], Theorem 1.2). Let $\mathbf{S}$ be a high-density subset of the squares, and suppose that $\mathbf{A}\subseteq\mathbb{N}$ has a complementary density growth exponent smaller than $1$. Then, whenever $\varepsilon>0$ and the natural number $N$ is sufficiently large in terms of $\varepsilon$, one has Lemma 2 (see [14], Theorem 2.2). Let $\mathbf{L}, \mathbf{M}$ and $\mathbf{N}$ be unions of arithmetic progressions modulo $q$, for some natural number $q$, and suppose that $\mathbf{N}\subseteq\mathbf{L}+\mathbf{M}$. Suppose also that $\mathbf{S}$ is a high-density subset of the squares relative to $\mathbf{L}$ and $\mathbf{A}\subseteq\mathbb{N}$ has an $\mathbf{M}$-complementary density growth exponent smaller than $1$. Then, whenever $\varepsilon>0$ and the natural number $N$ is sufficiently large in terms of $\varepsilon$, one has Lemma 3 (see [14], Theorem 1.3, (a)). Let $\mathbf{C}$ be a high-density subset of the cubes, and suppose that $\mathbf{A}\subseteq\mathbb{N}$ has a complementary density growth exponent smaller than $\theta$ for some positive number $\theta$. Then, whenever $\varepsilon>0$ and the natural number $N$ is sufficiently large in terms of $\varepsilon$, one has In order to prove Theorem 1, first we apply the Hardy-Littlewood method to study the problem Let $N$ be a sufficiently large positive integer, and let
$$
\begin{equation}
P=N^{3/20-2\varepsilon}\quad\text{and} \quad Q=N^{17/20+\varepsilon}.
\end{equation}
\tag{2.1}
$$
By Dirichlet’s lemma on rational approximation, each $\alpha\in[1/Q,1+1/Q]$ can be written in the form
$$
\begin{equation*}
\alpha=\frac{a}{q}+\lambda, \qquad |\lambda|\leqslant \frac{1}{qQ},
\end{equation*}
\notag
$$
for integers $a$ and $q$ such that $1\leqslant a\leqslant q\leqslant Q$ and $(a,q)=1$. Define
$$
\begin{equation}
\mathfrak{M}=\bigcup_{q\leqslant P}\bigcup_{\substack{1\leqslant a\leqslant q \\ (a,q)=1}}\mathfrak{M}(q,a) \quad\text{and}\quad \mathfrak{m} =\biggl[\frac{1}{Q},1+\frac{1}{Q}\biggr]\setminus\mathfrak{M},
\end{equation}
\tag{2.2}
$$
where
$$
\begin{equation*}
\mathfrak{M}(q,a)=\biggl[\frac{a}{q}-\frac{1}{qQ},\frac{a}{q}+\frac{1}{qQ}\biggr].
\end{equation*}
\notag
$$
For $k=2,3$ and $P_{k}=(N/16)^{1/k}$ we set
$$
\begin{equation}
f_{k}(\alpha)=\sum_{P_{k}<p\leqslant 2P_{k}}(\log p)e(p^{k}\alpha).
\end{equation}
\tag{2.3}
$$
Let
$$
\begin{equation*}
r(n)=\sum_{\substack{n=p_{1}^{2}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}\\ P_{2}<p_{1}\leqslant 2P_{2}\\ P_{3}<p_{2},p_{3},p_{4}\leqslant 2P_{3}}}\prod_{j=1}^{4}\log p_{j}.
\end{equation*}
\notag
$$
By orthogonality and (2.2), one has
$$
\begin{equation}
r(n)=\int_{0}^{1}f_{2}(\alpha)f_{3}^{3}(\alpha)e(-n\alpha)\,\mathrm{d}\alpha =\biggl\{\int_{\mathfrak{M}} +\int_{\mathfrak{m}}\biggr\}f_{2}(\alpha)f_{3}^{3}(\alpha)e(-n\alpha)\,\mathrm{d}\alpha.
\end{equation}
\tag{2.4}
$$
Set
$$
\begin{equation*}
C_{k}(\chi,a)=\sum_{h=1}^{q}\overline{\chi(h)}e\biggl(\frac{ah^{k}}{q}\biggr)\quad\text{and} \quad C_{k}(q,a)=C_{k}(\chi^{0},a),
\end{equation*}
\notag
$$
where $\chi\pmod q$ is a Dirichlet character and $k=2,3$. Let
$$
\begin{equation*}
B(n,q)=\sum_{\substack{1\leqslant a\leqslant q\\(a,q)=1}} C_{2}(q,a)C_{3}(q,a)C_{3}(q,a)C_{3}(q,a)e\biggl(-\frac{an}{q}\biggr)
\end{equation*}
\notag
$$
and
$$
\begin{equation}
A(n,q)=\frac{B(n,q)}{\varphi^{4}(q)}, \qquad \mathfrak{S}(n)=\sum_{q=1}^{\infty}A(n,q).
\end{equation}
\tag{2.5}
$$
Proposition 1. Let $P$, $Q$ and $\mathfrak{M}$ be defined in (2.1) and (2.2), respectively. Then for $n\in[N/4,N]$ and any $A>0$, The proof of Proposition 1 is a standard application of the iterative argument developed by Liu and Zhan (see [16], [17] and other papers). Thus we omit its proof here. We set
$$
\begin{equation*}
\begin{gathered} \, \mathscr{E}_{*}=\bigl\{n\in\mathbb{N}\colon n\equiv0\ (\operatorname{mod} 2),\ n\neq p_{1}^{2}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}\bigr\}, \\ \mathscr{E}_{**}=\bigl\{n\in\mathbb{N}\colon n\equiv1\ (\operatorname{mod} 2),\ n\neq p_{1}^{2}+p_{2}^{2}+p_{3}^{3}+p_{4}^{3}+p_{5}^{3}\bigr\} \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation}
E_{*}(N)=|\mathscr{E}_{*}|_{0}^{N}, \qquad E_{**}(N)=|\mathscr{E}_{**}|_{0}^{N}.
\end{equation}
\tag{2.9}
$$
Proposition 2. Let $E_{*}(N)$ be defined in (2.9). Then We prove this proposition in § 4. Proof of Theorem 1. Let the integers $N_{j}$ for $j>0$ be defined by the following recursive formula where $\lceil N \rceil$ denotes the least integer not smaller than $N$. In addition, let $J$ be the least positive integer satisfying $N_{J}=2$; then $J=O(L)$.
We define
$$
\begin{equation*}
\begin{gathered} \, \mathbf{A}_{1}=\bigl\{p_{1}^{2}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}\bigr\}, \qquad \mathbf{S}_{1}=\{p^{2}\}, \qquad \mathbf{L}_{1}=\bigl\{n\in\mathbb{N}\colon n\equiv1\ (\operatorname{mod}{24})\bigr\}, \\ \mathbf{M}_{1}=\bigl\{n\in\mathbb{N}\colon n\equiv0\ (\operatorname{mod} 2)\bigr\} \quad\text{and}\quad \mathbf{N}_{1}=\bigl\{n\in\mathbb{N}\colon n\equiv1\ (\operatorname{mod} 2)\bigr\}. \end{gathered}
\end{equation*}
\notag
$$
It is easy to see that $\mathbf{L}_{1}$ is a union of arithmetic progressions modulo $24$, and $\mathbf{M}_{1}$ and $\mathbf{N}_{1}$ are unions of arithmetic progressions modulo $2$ which satisfy $\mathbf{N}_{1}\subseteq\mathbf{L}_{1}+\mathbf{M}_{1}$. Then, we deduce from the Prime Number Theorem for arithmetic progressions that
$$
\begin{equation*}
\langle\mathbf{S}_{1}\wedge\mathbf{L}_{1}\rangle_{0}^{N}\gg N^{1/2}L^{-1};
\end{equation*}
\notag
$$
hence $\mathbf{S}_{1}$ is a high-density subset of the squares relative to $\mathbf{L}_{1}$. Using Proposition 2 we obtain
$$
\begin{equation*}
|\overline{\mathbf{A}_{1}}\cap\mathbf{M}_{1}|_{0}^{N} =E_{*}(N)\ll N^{1-1/12+\varepsilon},
\end{equation*}
\notag
$$
hence $\mathbf{A}_{1}$ has $\mathbf{M}_{1}$-complementary density growth exponent smaller than $1$. By Lemma 2
$$
\begin{equation*}
|\mathscr{E}_{**}|_{2N}^{3N}\ll N^{\varepsilon-1/2}|\mathscr{E}_{*}|_{N}^{3N} \ll N^{\varepsilon-1/2}E_{*}(3N)\ll N^{1/2-1/12+\varepsilon}.
\end{equation*}
\notag
$$
It follows from (2.10) that
$$
\begin{equation}
E_{**}(N)\leqslant 3+\sum_{j=1}^{J}|\mathscr{E}_{**}|_{2N_{j}}^{3N_{j}} \ll N^{1/2-1/12+\varepsilon}.
\end{equation}
\tag{2.11}
$$
Similarly, let
$$
\begin{equation*}
\mathbf{A}_{2}=\bigl\{p_{1}^{2}+p_{2}^{2}+p_{3}^{3}+p_{4}^{3}+p_{5}^{3}\bigr\} \quad\text{and}\quad \mathbf{C}_{2} =\{p^{3}\}.
\end{equation*}
\notag
$$
Then we deduce from the Prime Number Theorem that hence $\mathbf{C}_{2}$ is a high-density subset of the cubes. By (2.11) we obtain
$$
\begin{equation*}
|\overline{\mathbf{A}_{2}}|_{0}^{N}\ll N^{1/2-1/12+\varepsilon},
\end{equation*}
\notag
$$
thus, $\mathbf{A}_{2}$ has a complementary density growth exponent smaller than $1/2$. By Lemma 3 and (1.4),
$$
\begin{equation*}
\begin{aligned} \, |\mathscr{E}|_{2N}^{3N}&\ll N^{\varepsilon-1/3}|\mathscr{E}_{**}|_{2N}^{3N} +N^{\varepsilon-1}(|\mathscr{E}_{**}|_{N}^{3N})^{2} \\ &\ll N^{\varepsilon-1/3}E_{**}(3N)+N^{\varepsilon-1}E_{**}^{2}(3N) \ll N^{1/6-1/12+\varepsilon}. \end{aligned}
\end{equation*}
\notag
$$
Consequently, we deduce from (2.10) that
$$
\begin{equation*}
E(N)\leqslant 3+\sum_{j=1}^{J}|\mathscr{E}|_{2N_{j}}^{3N_{j}}\ll N^{1/6-1/12 +\varepsilon},
\end{equation*}
\notag
$$
which implies Theorem 1. § 3. Some lemmas Lemma 4 (see [18], Theorem 1.1). Suppose that $\alpha$ is a real number and where Lemma 5. Suppose that $\alpha$ is a real number and there exist integers $a\in\mathbb{Z}$ and $q\in\mathbb{N}$ such that The proof of the upper bound for $f_{2}(\alpha)$ and $f_{3}(\alpha)$ can be found in Theorem 3 of Kumchev [19] and Lemma 2.3 of Zhao [20], respectively. We define
$$
\begin{equation*}
\mathfrak{N}(q,a)=\biggl[\frac{a}{q}-\frac{1}{qN^{5/6}},\,\frac{a}{q}+\frac{1}{qN^{5/6}}\biggr] \quad\text{and}\quad \mathfrak{N}=\bigcup_{q\leqslant N^{1/6}}\bigcup_{\substack{1\leqslant a\leqslant q\\ (a,q)=1}}\mathfrak{N}(q,a).
\end{equation*}
\notag
$$
If we write
$$
\begin{equation*}
\mathfrak{m}_{1}=\mathfrak{m}\cap\mathfrak{N} \quad\text{and}\quad \mathfrak{m}_{2}=\mathfrak{m}\setminus\mathfrak{N},
\end{equation*}
\notag
$$
then Proof. If $\alpha\in \mathfrak{m}_{1}$, then
$$
\begin{equation*}
1 \leqslant a \leqslant q \leqslant N^{1/6}, \qquad |q \alpha-a| \leqslant N^{-5/6}\quad\text{and} \quad (a,q)=1.
\end{equation*}
\notag
$$
Since $\alpha\notin \mathfrak{M}$, we have either $q>P$ or $|q\alpha-a|>Q^{-1}$. We use Lemma 4 and obtain the upper bound for $f_{3}(\alpha)$. The lemma is proved. Proof. One has
$$
\begin{equation*}
1 \leqslant a \leqslant q \leqslant N^{1/2}, \qquad|q \alpha-a| \leqslant N^{-1/2}\quad\text{and} \quad(a, q)=1.
\end{equation*}
\notag
$$
Since $\alpha\in\mathfrak{m}_{2}=\mathfrak{m}\setminus\mathfrak{N}$, we have either $q>N^{1/6}$ or $N|q\alpha-a|>N^{1/6}$. We use Lemma 5 and obtain the upper bounds for $f_{2}(\alpha)$ and $f_{3}(\alpha)$. The lemma is proved. We can deduce this from Hua’s lemma. We can also find this result in Lemma 2.6 of Zhao [21]. Lemma 9 (see [15], Lemma 3.1). For $k\geqslant 3$ let $\mathscr{M}$ be the union of intervals $\mathscr{M}(q,a)$ for Lemma 10. For $\gamma\in\mathbb{R}$ set We obtain this lemma by taking $k=3$ and $P=Q=P_{3}$ in Lemma 2.2 of Zhao [15]. § 4. Proof of Proposition 2In this section we give the proof of Proposition 2. Let
$$
\begin{equation*}
U(-\alpha)=\sum_{n\in(\mathscr{E}_{*})_{0}^{N}}e(-n\alpha).
\end{equation*}
\notag
$$
Noting that $r(n)=0$ for all $n\in(\mathscr{E}_{*})_{0}^{N}$, by (2.4) one has
$$
\begin{equation}
\begin{aligned} \, \notag 0&=\sum_{n\in(\mathscr{E}_{*})_{0}^{N}}r(n) =\sum_{n\in(\mathscr{E}_{*})_{0}^{N}}\int_{0}^{1}f_{2}(\alpha)f_{3}^{3}(\alpha) e(-n\alpha)\,\mathrm{d}\alpha \\ &=\biggl\{\int_{\mathfrak{M}}+\int_{\mathfrak{m}}\biggr\}f_{2}(\alpha)f_{3}^{3}(\alpha) U(-\alpha)\,\mathrm{d}\alpha. \end{aligned}
\end{equation}
\tag{4.1}
$$
We use Proposition 1 and obtain
$$
\begin{equation*}
\begin{aligned} \, &\int_{\mathfrak{M}}f_{2}(\alpha)f_{3}^{3}(\alpha)U(-\alpha)\,\mathrm{d}\alpha =\sum_{n\in(\mathscr{E}_{*})_{0}^{N}} \int_{\mathfrak{M}}f_{2}(\alpha)f_{3}^{3}(\alpha)e(-n\alpha)\,\mathrm{d}\alpha \\ &\qquad =\sum_{n\in(\mathscr{E}_{*})_{0}^{N}}\biggl\{\frac{1}{54}\mathfrak{S}(n)\mathfrak{J}(n) +O(N^{1/2}L^{-A})\biggr\} \gg E_{*}(N)N^{1/2-\varepsilon}. \end{aligned}
\end{equation*}
\notag
$$
This, in combination with (4.1), gives
$$
\begin{equation}
\biggl|\int_{\mathfrak{m}}f_{2}(\alpha)f_{3}^{3}(\alpha) U(-\alpha)\,\mathrm{d}\alpha\biggr|\gg E_{*}(N)N^{1/2-\varepsilon}.
\end{equation}
\tag{4.2}
$$
On the other hand, by Cauchy’s inequality
$$
\begin{equation}
\begin{aligned} \, \notag \biggl|\int_{\mathfrak{m}}f_{2}(\alpha)f_{3}^{3}(\alpha)U(-\alpha)\,\mathrm{d}\alpha\biggr| &\ll\biggl(\int_{\mathfrak{m}}|f_{2}^{2}(\alpha)f_{3}^{6}(\alpha)| \,\mathrm{d}\alpha\biggr)^{1/2} \biggl(\int_{\mathfrak{m}}|U(-\alpha)|^{2}\,\mathrm{d}\alpha\biggr)^{1/2} \\ &\ll \biggl(\int_{\mathfrak{m}}|f_{2}^{2}(\alpha)f_{3}^{6}(\alpha)| \,\mathrm{d}\alpha\biggr)^{1/2}E_{*}^{1/2}(N). \end{aligned}
\end{equation}
\tag{4.3}
$$
Let
$$
\begin{equation*}
\mathscr{J}(t)=\int_{\mathfrak{m}_{2}}|f_{2}^{2}(\alpha)f_{3}^{t}(\alpha)|\,\mathrm{d}\alpha, \qquad 1\leqslant t\leqslant 6.
\end{equation*}
\notag
$$
Taking
$$
\begin{equation*}
g(\alpha)=f_{3}(\alpha), \qquad h(\alpha)=f_{3}(-\alpha)\quad\text{and} \quad G(\alpha)=|f_{2}^{2}(\alpha)f_{3}^{4}(\alpha)|
\end{equation*}
\notag
$$
in Lemma 9 we obtain
$$
\begin{equation}
\begin{aligned} \, \notag \mathscr{J}(6) &=N^{1/3}\mathscr{J}_{0}^{1/4} \biggl(\int_{\mathfrak{m}_{2}}|f_{2}^{4}(\alpha)f_{3}^{8}(\alpha)|\, \mathrm{d}\alpha\biggr)^{1/4}\mathscr{J}^{1/2}(5) +N^{7/24+\varepsilon}\mathscr{J}(5) \\ &=:H_{1}+H_{2}, \end{aligned}
\end{equation}
\tag{4.4}
$$
where
$$
\begin{equation*}
\mathscr{J}_{0}=\sup_{\beta\in[0,1)}\sum_{q\leqslant P_{3}^{3/4}} \sum_{\substack{1\leqslant a\leqslant q\\ (a,q)=1}}\int_{\mathscr{M}(q,a)} \frac{\omega_{3}^{2}(q)|h^{2}(\alpha+\beta)|}{(1+P_{3}^{3}|\alpha-a/q|)^{2}}\,\mathrm{d}\alpha
\end{equation*}
\notag
$$
for
$$
\begin{equation*}
\mathscr{M}(q,a)=\bigl\{\alpha\colon |q\alpha-a|\leqslant P_{3}^{-9/4}\bigr\}.
\end{equation*}
\notag
$$
We use Lemma 10 and obtain
$$
\begin{equation}
\mathscr{J}_{0}\ll \mathcal{L}(\gamma)\ll P_{3}^{2}N^{-1+\varepsilon} \ll N^{-1/3+\varepsilon}.
\end{equation}
\tag{4.5}
$$
For $\mathscr{I}(5)$, by Cauchy’s inequality and Lemma 8 we have
$$
\begin{equation}
\mathscr{J}(5)\leqslant \mathscr{J}^{1/2}(6) \biggl(\int_{\mathfrak{m}_{2}}|f_{2}^{2}(\alpha)f_{3}^{4}(\alpha)| \,\mathrm{d}\alpha\biggr)^{1/2}\ll N^{2/3+\varepsilon}\mathscr{J}^{1/2}(6).
\end{equation}
\tag{4.6}
$$
For the integral in (4.4) we use Lemma 7 and obtain
$$
\begin{equation}
\int_{\mathfrak{m}_{2}}|f_{2}^{4}(\alpha)f_{3}^{8}(\alpha)|\,\mathrm{d}\alpha \ll \mathscr{J}(6)\sup_{\alpha\in\mathfrak{m}_{2}}f_{2}^{2}(\alpha)f_{3}^{2}(\alpha) \ll N^{107/72+\varepsilon}\mathscr{J}(6).
\end{equation}
\tag{4.7}
$$
We deduce from (4.5)–(4.7) that
$$
\begin{equation}
H_{1}\ll N^{275/288+\varepsilon}\mathscr{J}^{1/2}(6).
\end{equation}
\tag{4.8}
$$
It follows from (4.6) that Substituting (4.8) and (4.9) into (4.4) one has
$$
\begin{equation*}
\mathscr{J}(6)\ll N^{{275}/{288}+\varepsilon}\mathscr{J}^{1/2}(6) +N^{{23}/{24}+\varepsilon}\mathscr{J}^{1/2}(6).
\end{equation*}
\notag
$$
Therefore, Applying Lemmas 6 and 8 to estimate $\displaystyle \int_{\mathfrak{m}_{1}}|f_{2}^{2}(\alpha)f_{3}^{6}(\alpha)|\,\mathrm{d}\alpha$ we obtain
$$
\begin{equation}
\int_{\mathfrak{m}_{1}}|f_{2}^{2}(\alpha)f_{3}^{6}(\alpha)|\,\mathrm{d}\alpha \ll\sup_{\alpha\in\mathfrak{m}_{1}}|f_{3}^{2}(\alpha)| \int_{0}^{1}|f_{2}^{2}(\alpha)f_{3}^{4}(\alpha)|\,\mathrm{d}\alpha \ll N^{{28}/{15}+\varepsilon}.
\end{equation}
\tag{4.11}
$$
We deduce from (3.2), (4.10) and (4.11) that
$$
\begin{equation}
\int_{\mathfrak{m}}|f_{2}^{2}(\alpha)f_{3}^{6}(\alpha)|\,\mathrm{d}\alpha \ll N^{{23}/{12}+\varepsilon}.
\end{equation}
\tag{4.12}
$$
Substituting (4.2) and (4.12) into (4.3) we have
$$
\begin{equation*}
\begin{aligned} \, E_{*}(N)N^{1/2-\varepsilon}\ll (N^{{23}/{12} +\varepsilon})^{1/2}E_{*}^{1/2}(N), \end{aligned}
\end{equation*}
\notag
$$
which implies that This completes the proof of Proposition 2.
§ 5. Sketch of the proof of Theorem 2Similarly to the proof of Theorem 1, we also prove Theorem 2. First we consider the cubic Waring-Goldbach problem Set
$$
\begin{equation*}
\begin{aligned} \, \mathscr{E}_{2} &=\bigl\{n\in\mathbb{N}\colon n\equiv 1\ (\operatorname{mod} 2),n\not\equiv 0, \pm2\ (\operatorname{mod} 9), n\not\equiv 0\ (\operatorname{mod} 7), \\ &\qquad n\neq p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}+p_{5}^{3}\bigr\}, \end{aligned}
\end{equation*}
\notag
$$
and let $E_{2}(N)$ denote the cardinality of $\mathscr{E}_{2}\cap(0,N]$. Zhao [15] proved that Let
$$
\begin{equation*}
\mathbf{A}=\bigl\{p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}+p_{5}^{3}\bigr\}, \qquad \mathbf{S}=\{p^{2}\}.
\end{equation*}
\notag
$$
Similarly to the proof of Theorem 1 we obtain
$$
\begin{equation*}
|\overline{\mathbf{A}}|_{0}^{N}\ll N^{1-1/12+\varepsilon}.
\end{equation*}
\notag
$$
By Lemma 1,
$$
\begin{equation*}
|\mathscr{E}_1|_{2N}^{3N}\ll N^{1/2-1/12+\varepsilon},
\end{equation*}
\notag
$$
where $\mathscr{E}_1$ was defined in (1.5). It follows from (2.10) that
$$
\begin{equation*}
E_{1}(N)\ll \sum_{j=1}^{J}|\mathscr{E}|_{2N_{j}}^{3N_{j}} \ll N^{1/2-1/12+\varepsilon}.
\end{equation*}
\notag
$$
This completes the proof of Theorem 2. AcknowledgementThe author would like to thank the referees for many useful comments made. 
Citation in format AMSBIB
\Bibitem{HanLiu23} |
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