| Sbornik: Mathematics |
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Slim exceptional sets of Waring–Goldbach problem: two squares, two cubes and two biquadrates Sh. Tian Department of Mathematics, Tongji University, Shanghai, P. R. China
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    § 1. IntroductionLet $n,k_1,k_2,\dots,k_s$ be natural numbers such that $2\leqslant k_1\leqslant k_2\leqslant \dots \leqslant k_s$, and let $n > s$. The Waring problem of mixed type concerns the representation of the natural number $n$ in the form Not very much is known on results of this kind. For references on this subject we refer to § P12 of LeVeque [6] and the bibliography of Vaughan [8].In 1970 Vaughan [7] obtained an asymptotic formula for the number of representations of an integer as a sum of two squares, two cubes and two biquadrates. Let $R(n)$ denote the number of representations of the integer $n$ in the form where $x_i \in \mathbb{N} (1\leqslant i \leqslant 6)$; Vaughan proved that
$$
\begin{equation*}
R(n)=\frac{\Gamma^2(3/2)\Gamma^2(4/3)\Gamma^2(5/4)}{\Gamma(13/6)} \mathfrak{S}_{2,3,4}(n)n^{7/6}+O(n^{7/6-1/96+\varepsilon}),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\mathfrak{S}_{2,3,4}(n)=\sum_{q=1}^{\infty}\frac{1}{q^6} \sum_{\substack{a=1\\(a,q)=1}}^{q}\prod_{i=1}^3 \biggl(\sum_{x_i=1}^{q}e\biggl(\frac{ax_i^{i+1}}{q}\biggr)\biggr)^2 e\biggl(-\frac{an}{q}\biggr).
\end{equation*}
\notag
$$
It is reasonable to conjecture that a sufficiently large even integer $n$ can be expressed as a sum of two squares, two cubes and two biquadrates of prime numbers. That is, for a sufficiently large even integer $n$ the equation is solvable in primes $p_j$ ($1 \leqslant j \leqslant 6$). (Here the letter $p$, with or without subscript, always means a prime number.) This conjecture is perhaps out of reach at present. However, it is possible to obtain a weaker result with $p_j$ replaced by an almost prime. In 2015 Lü [5] proved that for every sufficiently large even integer $n$ the equation is solvable with $x$ being an almost prime $\mathcal{P}_6$ and $p_j$ ($j=2,3,4,5,6$) being primes, where $\mathcal{P}_j$ denotes an almost prime with at most $j$ prime factors, counted according to multiplicity. In 2018 Liu [4] enhanced this result and showed that (1.2) is solvable with $x$ being an almost prime $\mathcal{P}_4$ and $p_j$ being primes. On the other hand, in 2022 Zhu [13] proved that every sufficiently large even integer $n$ can be represented as two squares of primes, two cubes of primes, two biquadrates of primes and 17 powers of 2, that is,
$$
\begin{equation*}
n=p_1^2+p_2^2+p_3^3+p_4^3+p_5^4+p_6^4+2^{v_1}+\dots +2^{v_{17}}.
\end{equation*}
\notag
$$
Let $E(N)$ denote the number of positive even integers $n\in (N/2,N]$ which cannot be represented as (1.1). In 2018 Zhang and Li [10] considered the exceptional set of problem (1.1) and obtained and in 2021 they [11] improved this result and obtainedIn this paper we continue to sharpen the above result and establish the following result. Theorem. Let $\mathcal{E}$ denote the set of integers which cannot be represented as a sum of two squares, two cubes and two biquadrates of primes, and for $n \leqslant N$ let ${E(N)=|\mathcal{E}|}$. Then for any $\varepsilon> 0$ we have We establish the theorem by applying the Hardy–Littlewood method. Our improvement benefits from the works of Kawada and Wooley [2] and Zhao [12]. Unlike the treatment in [11], we will establish a result on the exceptional set in a related Waring–Goldbach problem with fewer summands. Then, with the help of arguments from [2], we can obtain a better result. NotationIn this paper $N$ always denotes a sufficiently large even integer. Let $\varepsilon\in(0,10^{-10})$. The constants in $O$-terms and $\ll$-symbols depend at most on $\varepsilon$. As usual, we use $\varphi(n)$ and $d(n)$ to denote the Euler function and divisor function. We use $e(\alpha)$ to denote $e^{2\pi i\alpha}$. For a set $\mathcal{A}$, $|\mathcal{A}|$ denotes its cardinality. § 2. Preliminary and outline of methodIn order to better explain Lemmas 2.1 and 2.2, we need to introduce some notation. When $\mathcal{C} \subseteq \mathbb{N}$, we write $\overline{\mathcal{C}}$ for the complement $\mathbb{N}\setminus \mathcal{C}$ of $\mathcal{C}$ within $\mathbb{N}$. When $a$ and $b$ are nonnegative integers, it is convenient to denote by $(\mathcal{C})_a^b$ the set $\mathcal{C}\cap(a,b]$ and by $|\mathcal{C}|_a^b$ the cardinality of $\mathcal{C}\cap(a,b]$. Next, for $\mathcal{C}, \mathcal{D} \subseteq \mathbb{N}$ we define
$$
\begin{equation*}
\mathcal{C}\pm\mathcal{D}=\{c \pm d\colon c \in \mathcal{C}\text{ and } d \in \mathcal{D} \}.
\end{equation*}
\notag
$$
When $k\in \mathbb{N}$, a subset $\mathcal{Q}$ of $\mathbb{N}$ is a high-density subset of the $k$th powers if (i) $\mathcal{Q} \subseteq \{n^k,\, n \in \mathbb{N}\}$ and (ii) $|\mathcal{Q}|_0^N > N^{1/k-\varepsilon}$. For $\theta > 0$ a set $\mathcal{R}\subseteq \mathbb{N}$ is said to have complementary density growth exponent smaller than $\theta$ if there exists a positive number $\delta$ such that $|\overline{\mathcal{R}}|< N^{\theta-\delta}$. For $q \in \mathbb{N}$ and $a \in \{0,1,\dots,q-1\}$, let $\mathcal{P}_a=\mathcal{P}_{a,q}$ denote We say that a set $\mathcal{L}$ is a union of arithmetic progressions modulo $q$ if
$$
\begin{equation*}
\mathcal{L}=\bigcup_{\mathbf{l}\in\mathfrak{L}}P_{\mathbf{l},q}
\end{equation*}
\notag
$$
for some subset $\mathfrak{L}$ of $\{ 0,1,\dots,q-1 \}$. And it is convenient to write
$$
\begin{equation*}
\langle\mathcal{C}\wedge\mathcal{L}\rangle_a^b =\min_{\mathbf l\in\mathfrak{L}}|\mathcal{C}\cap\mathcal{P}_{\mathbf{l},q}|_a^b,
\end{equation*}
\notag
$$
where $\mathcal{C} \subseteq \mathbb{N}$ and $a,b \in \mathbb{Z}$. When $k \in \mathbb{N}$ and $\mathcal{L}$ is a union of arithmetic progressions modulo $q$, a subset $\mathcal{Q}$ of $\mathcal{N}$ is a high-density subset of the $k$th powers relative to $\mathcal{L}$ if (i) $\mathcal{Q} \subseteq \{n^k, n\in \mathbb{N}\}$ and (ii) $\langle\mathcal{Q}\wedge \mathcal{L}\rangle_0^N \gg_q N^{1/k-\varepsilon}$. For $\theta > 0$, a set $\mathcal{R} \subseteq \mathbb{N}$ is said to have $\mathcal{L}$-complementary density growth exponent smaller than $\theta$ if $|\overline{\mathcal{R}} \cap \mathcal{L}|_0^N < N^{\theta-\delta}$. Lemma 2.1. Let $\mathcal{L}$, $\mathcal{M}$ and $\mathcal{N}$ be unions of arithmetic progressions modulo $q$, for some natural number $q$, and suppose that $\mathcal{N} \subseteq \mathcal{L}+\mathcal{M}$. Suppose also that $\mathcal{S}$ is a high-density subset of the squares relative to $\mathcal{L}$, and that $\mathcal{A} \subseteq \mathbb{N}$ has $\mathcal{M}$-complementary density growth exponent smaller than $1$. Then, whenever $\varepsilon >0$ and $N$ is a natural number sufficiently large in terms of $\varepsilon$, we have Lemma 2.2. Let $\mathcal{C}$ be a high-density subset of the cubes, and suppose that ${\mathcal{A} \subseteq \mathbb{N}}$ has complementary density growth exponent smaller than $\theta$ for some positive number $\theta$. Then, whenever $\varepsilon > 0$ and $N$ is a natural number sufficiently large in terms of $\varepsilon$, we have The proof follows from Theorems 1.2, 2.2 and 4.1, (a), in [2]. In order to prove the theorem, we apply first the Hardy–Littlewood method to study the problem where $N$ is a sufficiently large positive integer.For this purpose we set $A$ to be a large absolute constant which is sufficiently large in terms of the constants $c$ and $c^*$ in Lemmas 3.3 and 3.4,
$$
\begin{equation*}
Q_0=\log^{A} N, \qquad Q_1=N^{1/6} \quad\text{and}\quad Q_2=N^{5/6},
\end{equation*}
\notag
$$
and for $(a,q)=1$, $0<a<q$, we put
$$
\begin{equation*}
\begin{gathered} \, \mathfrak M_0(q,a)=\biggl(\frac{a}{q}-\frac{Q_0}{N},\frac{a}{q}+\frac{Q_0}{N}\biggr], \qquad \mathfrak M(q,a)=\biggl(\frac{a}{q}-\frac{1}{qQ_2},\frac{a}{q}+\frac{1}{qQ_2}\biggr], \\ \mathfrak M_0=\bigcup_{q\leqslant Q_0} \bigcup_{\substack{a=1\\(a,q)=1}}^{q}{\mathfrak M_0(q,a)}, \qquad \mathfrak M=\bigcup_{q\leqslant Q_1} \bigcup_{\substack{a=1\\(a,q)=1}}^{q}{\mathfrak M(q,a)}, \\ \mathfrak J=\biggl(-\frac{1}{Q_2},1-\frac{1}{Q_2}\biggr], \qquad \mathfrak m_1=\mathfrak J\setminus \mathfrak M, \quad \mathfrak m_2=\mathfrak M\setminus \mathfrak M_0, \quad \mathfrak m=\mathfrak m_1\cup \mathfrak m_2. \end{gathered}
\end{equation*}
\notag
$$
Then we have the Farey dissection Let
$$
\begin{equation*}
P_k=\biggl[\frac{N^{1/k}}{2}\biggr]\quad\text{and} \quad f_k(\alpha)=\sum_{P_k<p\leqslant 2P_k}e(p^k\alpha)\log p,
\end{equation*}
\notag
$$
where $[N]$ is the least integer not smaller than $N$, and
$$
\begin{equation*}
\begin{aligned} \, \mathcal R(n) &=\int_{-1/Q_2}^{1-1/Q_2}{f_2(\alpha)f_3(\alpha)f_4^2(\alpha)e(-\alpha n)}\,d\alpha \\ &=\biggl\{\int_{\mathfrak{M}_0}+\int_{\mathfrak{m}} \biggr\} f_2(\alpha)f_3(\alpha)f_4^2(\alpha)e(-\alpha n)\,d\alpha. \end{aligned}
\end{equation*}
\notag
$$
We set
$$
\begin{equation*}
S_k(q,a)=\sum_{\substack{r=1\\(a,q)=1}}^q e\biggl(\frac{ar^k}{q}\biggr)
\end{equation*}
\notag
$$
for $k=2,3,4$. Let
$$
\begin{equation*}
A(n,q)=\frac{1}{\varphi^4(q)}S_2(q,a)S_3(q,a)S_4^2(q,a)e\biggl(-\frac{an}{q}\biggr) \quad\text{and}\quad \mathfrak S(n)=\sum_{q=1}^{\infty}A(q,n).
\end{equation*}
\notag
$$
Proposition 2.1. For $n \in [N/2,N]$ we have The proof of Proposition 2.1 is a standard application. Thus we omit it. We set
$$
\begin{equation*}
\begin{gathered} \, \mathcal{E}^*=\{n\in \mathbb{N}\colon n\equiv2,4 \ (\operatorname{mod} 6), \ n\neq p_1^2+p_2^3+p_3^4+p_4^4 \}, \\ \mathcal{E}^{**}=\{n\in \mathbb{N}\colon n\equiv3,5,9,11,15,17,21,23 \ (\operatorname{mod}24), \ n\neq p_1^2+p_2^2+p_3^3+p_4^4+p_5^4 \}, \\ E^*(N)=|\mathcal{E}^*|_0^N \quad\text{and}\quad E^{**}(N)=|\mathcal{E}^{**}|_0^N. \end{gathered}
\end{equation*}
\notag
$$
We prove this proposition in § 4. Proof of the theorem. Let the integers $N_j$ for $j>0$ be determined by the following recurrence formula:
$$
\begin{equation*}
N_0=\biggl[\frac{N}{2}\biggr]\quad\text{and} \quad N_{j+1}=\biggl[\frac{2N_j}{3}\biggr], \quad j\geqslant 0.
\end{equation*}
\notag
$$
Meanwhile, let $J$ be the least positive integer satisfying $N_{\mkern-1mu J}\!=\!2$, so that ${J\!=\!O(\log N)}$. We put
$$
\begin{equation*}
\begin{gathered} \, \mathcal{A}_1=\{p_1^2+p_2^3+p_3^4+p_4^4\}, \qquad \mathcal{S}=\{p^2\}, \qquad \mathcal{L}=\{n\in \mathbb{N} \colon n \equiv 1 \ (\operatorname{mod} 24)\}, \\ \mathcal{M} = \{n\in \mathbb{N}\colon n \equiv 0\ (\operatorname{mod}2) \} \quad\text{and}\quad \mathcal{N}=\{n\in \mathbb{N}\colon n\equiv 1\ (\operatorname{mod}2)\}. \end{gathered}
\end{equation*}
\notag
$$
Then we see that $\mathcal{N} \subseteq \mathcal{L}+\mathcal{M}$ and deduce from the prime number theorem in arithmetic progressions that
$$
\begin{equation*}
\langle\mathcal{S}\wedge\mathcal{L}\rangle_0^N \gg N^{1/2-\varepsilon}.
\end{equation*}
\notag
$$
From Proposition 2.2 we obtain
$$
\begin{equation*}
|\overline{\mathcal{A}_1}\cap\mathcal{M}|_0^N=E^*(N)\ll N^{89/96+\varepsilon}.
\end{equation*}
\notag
$$
Using Lemma 2.2 we have
$$
\begin{equation*}
|\mathcal{E}^{**}|_{2N}^{3N} \ll N^{\varepsilon-1/2}|\mathcal{E}^*|_N^{3N} \ll N^{\varepsilon-1/2}E^*(3N) \ll N^{89/96-1/2+\varepsilon}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
E^{**}(N) \leqslant 3+\sum_{j=1}^J |\mathcal{E}^{**}|_{2N_j}^{3N_j} \ll N^{89/96-1/2+\varepsilon}.
\end{equation*}
\notag
$$
Similarly, set
$$
\begin{equation*}
\mathcal{A}_2=\{p_1^2+p_2^2+p_3^3+p_4^4+p_5^4\}, \qquad \mathcal{C}=\{p^3\}.
\end{equation*}
\notag
$$
Then we deduce from the prime number theorem that From Lemma 2.2 we obtain
$$
\begin{equation*}
\begin{aligned} \, |\mathcal{E}|_{2N}^{3N} &\ll N^{\varepsilon-1/3}|\mathcal{E}^{**}|_{2N}^{3N} +N^{\varepsilon-1}(|\mathcal{E}^{**}|_N^{3N})^2 \\ &\ll N^{\varepsilon-1/3}E^{**}(3N)+N^{\varepsilon-1}(E^{**}(3N))^2 \ll N^{{41}/{96}-1/3+\varepsilon}. \end{aligned}
\end{equation*}
\notag
$$
Therefore, we have which implies the theorem. § 3. Some lemmasLemma 3.1. Let $2 \leqslant k_1 \leqslant k_2 \leqslant \dots \leqslant k_s$ be natural numbers such that Then Lemma 3.2. Let $Z(N)$ denote the subset of integers in the interval $[\frac12N, N]$, and let $Z= |Z(N)|$. Let $\xi\colon \mathbb{Z} \to \mathbb{C} $ be a function satisfying $|\xi(n)| \leqslant 1$, and set Then we have For a proof see (2.4) in [9]. Lemma 3.3. Let Lemma 3.4. Define the multiplicative function $\omega_k(q)$ by setting The proof can be found in Theorem 3 in [3] and Lemma 2.4 in [12] for $q > N^{1/6}$ or $|\alpha- a/q| > {1}/(qN^{5/6})$. Lemma 3.6. Let $\mathcal{M}$ be the union of the intervals $\mathcal{M}(q,a,P_3)$ for $1\leqslant a \leqslant q \leqslant P_3^{3/4}$ and $(a,q)=1$, where $\mathcal{M}(q,a,P_3)=\{ \alpha \colon | q\alpha-a| \leqslant P_3^{-9/4} \}$. Suppose that $G(\alpha)$ and $h(\alpha)$ are integrable functions of period $1$. Then we have § 4. Mean value estimatesLet
$$
\begin{equation*}
I_j=\int_{\mathfrak m_j}{|f_2(\alpha)f_3(\alpha)f_4^2(\alpha)K(\alpha)|}\,d\alpha, \qquad j=1,2,
\end{equation*}
\notag
$$
where $K(\alpha)$ is defined as in Lemma 3.2. Proposition 4.1. We have where $Z$ is defined as in Lemma 3.2. Proof. By Cauchy’s inequality we have
$$
\begin{equation*}
\begin{aligned} \, I_1 &\ll \biggl(\int_{\mathfrak m_1}|f_2^2(\alpha)f_3^2(\alpha)f_4^4(\alpha)|\, d\alpha \biggr)^{1/2}\biggl(\int_{\mathfrak{m_1}}|K(\alpha)|^2\, d\alpha \biggr)^{1/2} \\ &\ll \biggl(\int_{\mathfrak m_1}|f_2^2(\alpha)f_3^2(\alpha)f_4^4(\alpha)|\, d\alpha \biggr)^{1/2}Z^{1/2}. \end{aligned}
\end{equation*}
\notag
$$
Let
$$
\begin{equation*}
\mathcal{I}(t)= \int_{\mathfrak m_1}|f_2^2(\alpha)f_3^t(\alpha)f_4^4(\alpha)|\, d\alpha, \qquad 1 \leqslant t \leqslant 2.
\end{equation*}
\notag
$$
Applying Lemma 3.6 to $h(\alpha)=f_4(\alpha)$ and $G(\alpha)=f_2^2(\alpha)f_3(\alpha)f_4^3(\alpha)$ we obtain
$$
\begin{equation}
\begin{aligned} \, \mathcal{I}(2) &= \int_{\mathfrak m_1}{|f_3(\alpha)h(\alpha)G(\alpha)|} \,d\alpha \nonumber \\ &\ll P_3\mathcal{J}_0^{1/4}\biggl(\int_{\mathfrak m_1}{|G(\alpha)|}^2 \,d\alpha\biggr)^{1/4}{\mathcal{I}(1)^{1/2}}+P_3^{7/8+\varepsilon}\mathcal{I}(1). \end{aligned}
\end{equation}
\tag{4.1}
$$
From Cauchy’s inequality and Lemma 3.1 we get that
$$
\begin{equation}
\mathcal{I}(1) \ll \biggl(\int_0^1 |f_2^2(\alpha)f_4^4(\alpha)|\, d\alpha\biggr)^{1/2}\mathcal{I}(2)^{1/2} \ll N^{1/2+\varepsilon}\mathcal{I}(2)^{1/2}.
\end{equation}
\tag{4.2}
$$
We deduce from Lemma 3.4 that
$$
\begin{equation}
\begin{aligned} \, \mathcal{J}_0 &\ll \sup_{\beta \in [0,1)}\sum_{q\leqslant Q} \sum_{\substack{a=1\\(a,q)=1}}^q \int_{|\alpha-a/q| \leqslant N}{\frac{\omega_3^2(q){| f_4(\alpha+\beta)|}^2}{(1+P_3^3(\alpha-a/q))^2}}\,d\alpha \nonumber \\ &\ll \sup_{\beta \in [0,1)} \mathcal L(\beta) \ll N^{-1/2+\varepsilon}. \end{aligned}
\end{equation}
\tag{4.3}
$$
Using Lemma 3.5 we obtain
$$
\begin{equation}
\begin{aligned} \, \int_{\mathfrak m_1}{|G(\alpha)|}^2\, d\alpha &\ll \max_{\alpha \in \mathfrak m_1}{|f_2(\alpha)|}^2{|f_4(\alpha)|}^2 \int_{\mathfrak{m}_1}{| f_2^2(\alpha)f_3^2(\alpha)f_4^4(\alpha)|} \,d\alpha \nonumber \\ &\ll N^{7/8+23/48+\varepsilon}I(2) \ll N^{65/48+\varepsilon}I(2). \end{aligned}
\end{equation}
\tag{4.4}
$$
By combining (4.1)–(4.4) we obtain
$$
\begin{equation}
\begin{aligned} \, \mathcal{I}(2) &\ll P_3N^{-1/8+\varepsilon}N^{65/192+\varepsilon}I(2)^{1/4}N^{1/4+\varepsilon} \mathcal{I}(2)^{1/4}+P_3^{7/8+\varepsilon}N^{1/2+\varepsilon}\mathcal{I}(2)^{1/2} \nonumber \\ &\ll N^{{153}/{192}+\varepsilon}\mathcal{I}(2)^{1/2} +N^{{19}/{24}+\varepsilon} \mathcal{I}(2)^{1/2}, \nonumber \\ \mathcal{I}(2)^{1/2} &\ll N^{{153}/{192}+\varepsilon}. \end{aligned}
\end{equation}
\tag{4.5}
$$
Therefore,
$$
\begin{equation*}
I_1 \ll \biggl(\int_{\mathfrak m_1}|f_2^2(\alpha)f_3^2(\alpha)f_4^4(\alpha)| \,d\alpha \biggr)^{1/2}Z^{1/2} \ll \mathcal{I}(2)^{1/2}Z^{1/2} \ll N^{{153}/{192}+\varepsilon}Z^{1/2}.
\end{equation*}
\notag
$$
This completes Proposition 4.1. Proof. For $\alpha \in \mathfrak m_2$, by Lemma 3.3 and $Q=N^{1/6}$ we have
$$
\begin{equation}
|f_2(\alpha)| \ll N^{1/12}P_2^{11/20+\varepsilon}+\frac{P_2(\log N)^{c^*}}{q^{1/2-\varepsilon}(1+X^2|\lambda|)^{1/2}} \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\ll N^{43/120+\varepsilon}+\frac{N^{1/2}(\log N)^{c^*}}{q^{1/2-\varepsilon}(1+N|\lambda|)^{1/2}},
\end{equation}
\tag{4.6}
$$
$$
\begin{equation}
|f_3(\alpha)| \ll N^{1/12}P_3^{11/20+\varepsilon}+\frac{P_3(\log N)^{c^*}}{q^{1/2-\varepsilon}(1+X^3|\lambda|)^{1/2}} \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\ll N^{4/15+\varepsilon}+\frac{N^{1/3}(\log N)^{c^*}}{q^{1/2-\varepsilon}(1+N|\lambda|)^{1/2}}
\end{equation}
\tag{4.7}
$$
and
$$
\begin{equation}
\begin{aligned} \, |f_4(\alpha)| &\ll N^{1/12}P_4^{11/20+\varepsilon}+\frac{P_4(\log N)^{c^*}}{q^{1/2-\varepsilon}(1+X^4|\lambda|)^{1/2}} \nonumber \\ &\ll N^{53/240+\varepsilon}+\frac{N^{1/4}(\log N)^{c^*}}{q^{1/2-\varepsilon}(1+N|\lambda|)^{1/2}}, \end{aligned}
\end{equation}
\tag{4.8}
$$
where $c^*$ is a positive constant. We set
Then we obtain
$$
\begin{equation*}
\begin{aligned} \, I_2 &\ll N^{4/15+\varepsilon}\int_{\mathfrak m_2}{|f_2(\alpha)f_4^2(\alpha)K(\alpha)|}\,d\alpha + \int_{\mathfrak m_2}{|V_3(\alpha)f_2(\alpha)f_4^2(\alpha)K(\alpha)| }\,d\alpha \\ &\ll N^{4/15+\varepsilon}\int_{\mathfrak m_2}{|f_2(\alpha)f_4^2(\alpha)K(\alpha)|}\,d\alpha + N^{43/120+\varepsilon}\int_{\mathfrak m_2}{| V_3(\alpha)f_4^2(\alpha)K(\alpha)| }\,d\alpha \\ &\qquad+ \int_{\mathfrak m_2}{|V_2(\alpha)V_3(\alpha)f_4^2(\alpha)K(\alpha)| }\,d\alpha \\ &\ll N^{4/15+\varepsilon}\int_{\mathfrak m_2}{|f_2(\alpha)f_4^2(\alpha)K(\alpha)|}\,d\alpha + N^{43/120+\varepsilon}\int_{\mathfrak m_2}{| V_3(\alpha)f_4^2(\alpha)K(\alpha)| }\,d\alpha \\ &\qquad + N^{{53}/{120}+\varepsilon}\int_{\mathfrak m_2}{|V_2(\alpha)V_3(\alpha)K(\alpha)| }\,d\alpha +\int_{\mathfrak m_2}{|V_2(\alpha)V_3(\alpha)V_4^2(\alpha)K(\alpha)| }\,d\alpha \\ &\ll I_{21}+I_{22}+I_{23}+I_{24}. \end{aligned}
\end{equation*}
\notag
$$
By Cauchy’s inequality and Lemma 3.1 we have
$$
\begin{equation}
\begin{aligned} \, I_{21} &\ll N^{4/15+\varepsilon} \biggl(\int_0^1{|K(\alpha)|}^2 \,d\alpha\biggr)^{1/2} \biggl(\int_0^1{|f_2(\alpha)f_4^2(\alpha)|}^2\,d\alpha\biggr)^{1/2} \nonumber \\ &\ll N^{4/15+\varepsilon}Z^{1/2}N^{1/2+\varepsilon} \ll Z^{1/2}N^{23/30+\varepsilon}. \end{aligned}
\end{equation}
\tag{4.9}
$$
By Lemma 3.4 and ${1}/{q}\leqslant \omega_4^2(q)$,
$$
\begin{equation}
\begin{aligned} \, &\int_{\mathfrak m_2}{|V_3^2(\alpha)f_4^2(\alpha)|} \,d\alpha \nonumber \\ &\quad \ll N^{2/3+\varepsilon}\sum_{q\leqslant N^{1/6}}\sum_{\substack{a=1\\(a,q)=1}}^q \int_{|\alpha-a/q|\leqslant {1}/(qN^{5/6})}{\frac{(\log N)^{2c^*} {|\sum_{p\leqslant N^{1/4}}e(\alpha p^4)|}^2}{q(1+N|\alpha-a/q|)}}\,d\alpha \nonumber \\ &\quad\ll N^{2/3+\varepsilon}{(\log N)}^{2c^*}\sum_{q\leqslant N^{1/6}}\sum_{\substack{a=1\\(a,q)=1}}^q \int_{|\alpha-a/q|\leqslant 1/(qN^{5/6})}\!{\frac{\omega_4^2(q) {|\sum_{p\leqslant N^{1/4}}e(\alpha p^4)|}^2}{1+N|\alpha-a/q|}}\,d\alpha \nonumber \\ &\quad \ll N^{2/3+\varepsilon}{(\log N)}^{2c^*} \mathcal L(0) \ll N^{1/6+\varepsilon} \end{aligned}
\end{equation}
\tag{4.10}
$$
and
$$
\begin{equation}
\begin{aligned} \, &\int_{\mathfrak m_2}{|V_2(\alpha)V_3(\alpha)| }\,d\alpha \nonumber \\ &\quad\ll N^{5/6}(\log N)^{2c^*} \sum_{q\leqslant N^{1/6}}\sum_{\substack{a=1\\(a,q)=1}}^q\int_{|\alpha-{a}/{q}|\leqslant 1/(qN^{5/6})}{\frac{1}{q^{1-2\varepsilon}(1+N|\alpha-a/q|)}}\,d\alpha \nonumber \\ &\quad \ll N^{5/6}(\log N)^{2c^*}N^{-1+\varepsilon} \ll N^{-1/6+\varepsilon}. \end{aligned}
\end{equation}
\tag{4.11}
$$
Applying (4.9) and the trivial bound $|f_4(\alpha)| \leqslant N^{1/4}$, we have
$$
\begin{equation}
\begin{aligned} \, I_{22} &\ll N^{43/120+1/4+\varepsilon}\biggl(\int_{\mathfrak m_2}{| V_3^2(\alpha)f_4^2(\alpha)| }\,d\alpha\biggr)^{1/2}\biggl(\int_0^1 |K(\alpha)|^2 \,d\alpha \biggr)^{1/2} \nonumber \\ &\ll Z^{1/2}N^{83/120+\varepsilon} \end{aligned}
\end{equation}
\tag{4.12}
$$
and
$$
\begin{equation}
\begin{aligned} \, I_{23} &\ll N^{53/120+\varepsilon}Z\biggl(\int_{\mathfrak m_2}{|V_3(\alpha)V_2(\alpha)| }\,d\alpha\biggr) \ll ZN^{11/40+\varepsilon}. \end{aligned}
\end{equation}
\tag{4.13}
$$
Similarly to (4.11) we have
$$
\begin{equation}
\begin{aligned} \, &\int_{\mathfrak m_2}{|V_3(\alpha)V^2_4(\alpha)| }\,d\alpha \nonumber \\ &\quad \ll N^{5/6}(\log N)^{3c^*} \!\sum_{q\leqslant N^{1/6}}\sum_{\substack{a=1\\(a,q)=1}}^q\int_{|\alpha-{a}/{q}|\leqslant 1/(qN^{5/6})}{\frac{1}{q^{3/2-3\varepsilon}(1\,{+}\,N|\alpha\,{-}\,a/q|)^{3/2}}}\,d\alpha \nonumber \\ &\quad\ll N^{5/6}(\log N)^{3c^*}N^{-1} \ll N^{-1/6}(\log N)^{3c^*}. \end{aligned}
\end{equation}
\tag{4.14}
$$
Note that for $\alpha \in \mathfrak m_2$ we have $q\gg Q_0$ or $|\alpha-a/q| \gg {Q_0}/{N}$. Thus,
$$
\begin{equation}
\sup_{\alpha \in \mathfrak m_2}|V_2(\alpha)| \ll N^{1/2}Q_0^{-1/2}(\log N)^{c^*}.
\end{equation}
\tag{4.15}
$$
Then we can conclude from (4.13) and (4.14) that
$$
\begin{equation}
\begin{aligned} \, I_{24} &\ll Z\sup_{\alpha \in \mathfrak m_2}{|V_2(\alpha)|}\int_{\mathfrak m_2}{|V_3(\alpha)V^2_4(\alpha)| }\,d\alpha \nonumber \\ &\ll ZN^{1/2}Q_0^{-1/2}N^{-1/6}(\log N)^{4c^*} \ll ZN^{1/3}Q_0^{-1/3}. \end{aligned}
\end{equation}
\tag{4.16}
$$
A combination of (4.9), (4.12), (4.13) and (4.16) shows that
$$
\begin{equation*}
I_2 \ll ZN^{1/3}Q_0^{-1/3}+Z^{1/2}N^{23/30+\varepsilon}.
\end{equation*}
\notag
$$
This completes the proof of Proposition 4.2. Proof of Proposition 2.2. Let $\mathcal F(N)$ denote the set of integers $N/2 \leqslant n \leqslant N$ such that
$$
\begin{equation}
\mathcal R(n)=\int_{\mathfrak M_0}{f_2(\alpha)f_3(\alpha)f_4^2(\alpha)e(-\alpha n)}\,d\alpha + \int_{\mathfrak m}{f_2(\alpha)f_3(\alpha)f_4^2(\alpha)e(-\alpha n)}\,d\alpha.
\end{equation}
\tag{4.17}
$$
By Proposition 2.1, for $n \in \mathcal F(N)$ we obtain
$$
\begin{equation}
\biggl| \int_{\mathfrak m}{f_2(\alpha)f_3(\alpha)f_4^2(\alpha)e(-\alpha n)}\,d\alpha \biggr| \gg \frac{n^{1/3}}{\log^4 n}.
\end{equation}
\tag{4.18}
$$
We choose the complex number $\xi (n)$ satisfying $|\xi (n)| = 1$ and such that
$$
\begin{equation}
\xi (n)\int_{\mathfrak m}{f_2(\alpha)f_3(\alpha)f_4^2(\alpha)e(-\alpha n)}\,d\alpha=\biggl| \int_{\mathfrak m}{f_2(\alpha)f_3(\alpha)f_4^2(\alpha)e(-\alpha n)}\,d\alpha \biggr|.
\end{equation}
\tag{4.19}
$$
Set $Z(N)= |\mathcal F(N)|$. From (4.18) and (4.19) we obtain
$$
\begin{equation}
\begin{aligned} \, \frac{Z(N)N^{1/3}}{\log^4 N} &\ll \sum_{n \in \mathcal F(N)}{\frac{n^{1/3}}{\log^4 n}} \ll \int_{\mathfrak m}{f_2(\alpha)f_3(\alpha)f_4^2(\alpha)K(\alpha)}\,d\alpha \nonumber \\ &\ll \biggl(\int_{\mathfrak m_1}+\int_{\mathfrak m_2}\biggr){f_2(\alpha)f_3(\alpha)f_4^2(\alpha)K(\alpha)}\,d\alpha, \end{aligned}
\end{equation}
\tag{4.20}
$$
where
$$
\begin{equation*}
K(\alpha)=\sum_{n \in \mathcal F(N)}{\xi(n)e(-\alpha n)}.
\end{equation*}
\notag
$$
From (4.20) and Propositions 4.1 and 4.2 we obtain
$$
\begin{equation}
\frac{Z(N)N^{1/3}}{\log^4 N} \ll {Z(N)}^{1/2}N^{153/192+\varepsilon}+Z(N)N^{1/3}Q_0^{-1}.
\end{equation}
\tag{4.21}
$$
Therefore, and
$$
\begin{equation*}
\mathcal E(N) \ll N^{89/96+\varepsilon}+\sum_{1\leqslant 2^j \leqslant N^{1/12}}{Z\biggl(\frac{N}{2^j}\biggr)} \ll N^{89/96+\varepsilon}.
\end{equation*}
\notag
$$
Now the proof of Proposition 2.2 is complete. 
Citation in format AMSBIB
\Bibitem{Tia25} |
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