Let $ [\alpha] $ denote the integral part of the real number $ \alpha $, and let $ N $ be a sufficiently large integer. In this paper, we proved that for $ 1 < c < \frac{38}{29} $, almost all $ n\in(N, 2N] $ can be represented as $ [p_1^c]+[p_2^c]+[p_3^c]+[p_4^c] = n $, where $ p_1, p_2, p_3, p_4 $ are prime numbers.
Citation: Jing Huang, Wenguang Zhai, Deyu Zhang. On a Diophantine equation with four prime variables[J]. AIMS Mathematics, 2025, 10(6): 14488-14501. doi: 10.3934/math.2025652
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Let $ [\alpha] $ denote the integral part of the real number $ \alpha $, and let $ N $ be a sufficiently large integer. In this paper, we proved that for $ 1 < c < \frac{38}{29} $, almost all $ n\in(N, 2N] $ can be represented as $ [p_1^c]+[p_2^c]+[p_3^c]+[p_4^c] = n $, where $ p_1, p_2, p_3, p_4 $ are prime numbers.
The Diophantine equation is a classical problem in number theory. Let $ [\alpha] $ denote the integral part of the real number $ \alpha $, and let $ N $ be a sufficiently large integer. In 1933, Segal [1,2] firstly studied additive problems with non-integer degrees and proved that for $ c > 1 $ being not an integer, there exists $ k(c) > 0 $ such that the Diophantine equation
$ \begin{equation} [x_1^c]+[x_2^c]+\cdots+[x_k^c] = N \end{equation} $ | (1.1) |
is solvable for $ k > k(c) $. Later, Deshouillers [3] and Arkhilov and Zhitkov [4] improved the Segal's bound for $ k(c) $. Laporta [5] demonstrated in 1999 that the equation
$ \begin{equation} [p_1^c]+[p_2^c] = n \end{equation} $ | (1.2) |
is solvable in primes $ p_1 $, $ p_2 $ provided that $ 1 < c < \frac{17}{16} $ and $ N $ is sufficiently large. Recently, the range of $ c $ in (1.2) was enlarged to $ 1 < c < \frac{14}{11} $ by Zhu [6].
In 1995, Laporta and Tolev [7] considered the equation
$ \begin{equation} [p_1^c]+[p_2^c]+[p_3^c] = n \end{equation} $ | (1.3) |
with prime variables $ p_1, p_2, p_3 $. Denote the weighted number of solutions of the Eq (1.3) by
$ \begin{equation} \mathcal{R}(n) = \sum\limits_{[p_1^c]+[p_2^c]+[p_3^c] = n}(\log p_1)(\log p_2)(\log p_3), \end{equation} $ | (1.4) |
where $ N/2 < n\leq N $ and $ N $ is a sufficiently large integer. They established the following asymptotic formula
$ \begin{equation*} \label{1.4} \mathcal{R}(n) = \frac{\Gamma^3\left(1+\frac{1}{c}\right)}{\Gamma\left(\frac{3}{c}\right)}n^{\frac{3}{c}-1} +O\left(N^{\frac{3}{c}-1}\exp\Big(-\log^{\frac{1}{3}-\delta}N\Big)\right) \end{equation*} $ |
for any $ 0 < \delta < \frac{1}{3} $ and $ 1 < c < \frac{17}{16} $. Afterwards, the range of $ c $ was enlarged to $ 1 < c < \frac{12}{11} $ by Kumchev and Nedeva [8], to $ 1 < c < \frac{258}{235} $ by Zhai and Cao [9], to $ 1 < c < \frac{137}{119} $ by Cai [10], to $ 1 < c < \frac{3113}{2703} $ by Li and Zhang[11], and to $ 1 < c < \frac{3581}{3106} $ by Baker [12].
In this paper, we shall investigate the solvability of the following Diophantine equation
$ \begin{align} n = [p_1^c]+[p_2^c]+[p_3^c]+[p_4^c] \end{align} $ | (1.5) |
in prime variables $ p_1, p_2, p_3, p_4 $. Denote the weighted number of solutions of the above equation by
$ \begin{equation} R(n) = \sum\limits_{[p_1^c]+[p_2^c]+[p_3^c]+[p_4^c] = n}(\log p_1)(\log p_2)(\log p_3)(\log p_4) \end{equation} $ | (1.6) |
and establish the following theorem.
Theorem 1. Let $ N $ be a sufficiently large integer. Then for $ 1 < c < \frac{38}{29} $ and $ n\in(\frac{N}{2}, N] $ but
$ O(N\exp(-\log^{\frac{1}{5}}N)) $ |
exceptions, we have
$ \begin{equation} R(n) = \frac{\Gamma^4\left(1+\frac{1}{c}\right)}{\Gamma\left(\frac{4}{c}\right)}n^{\frac{4}{c}-1}+O\left(N^{\frac{4}{c}-1}\exp\left(-\log^{\frac{1}{4}}N\right)\right), \end{equation} $ | (1.7) |
where the implied constant in the $ O $-term depends only on $ c $.
Notation. Throughout the paper, we assume that $ 1 < c < \frac{38}{29} $. The symbol $ N $ always denotes a sufficiently large integer. Let $ \varepsilon\in\Big(0, 10^{-10}(\frac{38}{29}-c)\Big) $. Let $ p $, with or without subscripts, be reserved for a prime number. We denote the fractional part of $ x $ by $ \{x\} $ and the distance from $ x $ to the nearest integer by $ \|x\| $. Let
$ \begin{align*} &&P = N^{\frac{1}{c}}, \ \ \tau = P^{1-c-\varepsilon}, \ \ e(x) = e^{2\pi ix}, \ \ S(\alpha) = \sum\limits_{p\leq P}(\log p)e(\alpha [p^c]), \\ &&S(\alpha, X) = \sum\limits_{X < p\leq 2X}(\log p)e(\alpha [p^c]), \ \ T(\alpha, X) = \sum\limits_{X < n\leq2X}e([n^c]\alpha).\\ \end{align*} $ |
To prove Theorem 1, we need the following lemmas.
Lemma 2.1. [13, Lemma 5] Suppose that $ z_n $ is a sequence of complex numbers, then we have
$ \begin{align*} \left|\sum\limits_{N\leq n\leq 2N}z_n\right|^2\leq\left(1+\frac{N}{Q}\right)\sum\limits_{q = 0}^{Q}\left(1-\frac{q}{Q}\right){\rm Re}\left(\sum\limits_{N\leq n\leq 2N-q}\overline{z_n}z_{n+q}\right), \end{align*} $ |
where $ {\rm Re}(t) $ and $ \overline{t} $ denote the real part and the conjugate of the complex number $ t $, respectively.
Lemma 2.2. [14, (3.3.4)] Suppose that $ |x| > 0 $ and $ c > 1 $. Then for any exponent pair $ (\kappa, \lambda) $, $ M\leq a < b\leq2M $, we have
$ \begin{align*} \sum\limits_{a\leq n\leq b}e(xn^c)\ll(|x|M^c)^\kappa M^{\lambda-\kappa}+\frac{M^{1-c}}{|x|}. \end{align*} $ |
Lemma 2.3. [15, Lemma 12] Suppose that $ t $ is not an integer and $ H\geq3 $. Then for any $ \alpha\in(0, 1) $, we have
$ \begin{align*} e(-\alpha\{t\}) = \sum\limits_{|h|\leq H}c_h(\alpha)e(ht)+O\left(\min\bigg(1, \frac{1}{H\|t\|}\bigg)\right), \end{align*} $ |
where
$ \begin{align*} c_h(\alpha) = \frac{1-e(-\alpha)}{2\pi i(h+\alpha)}. \end{align*} $ |
Lemma 2.4. [13, Lemma 3] Suppose that $ 3 < U < V < Z < X $, and $ \{Z\} = \frac{1}{2}, \ X\geq64Z^2U, \ Z\geq 4U^2, \ V^3\geq32X $. We further assume that $ F(n) $ is a complex valued function such that $ |F(n)|\leq1 $. Then the sum
$ \begin{align*} \sum\limits_{X\leq n\leq2X}\Lambda(n)F(n) \end{align*} $ |
may be decomposed into $ O(\log^{10}X) $ sums, each of which either of type Ⅰ:
$ \sum\limits_{M\leq m\leq2M}a(m)\sum\limits_{N\leq n\leq2N}F(mn) $ |
with $ N > Z $, where $ a(m) \ll m^{\varepsilon} $ and $ X\ll MN\ll X $, or of type Ⅱ:
$ \sum\limits_{M\leq m\leq2M}a(m)\sum\limits_{N\leq n\leq2N}b(n)F(mn) $ |
with $ U\ll M\ll V $, where $ a(m) \ll m^{\varepsilon}, b(n) \ll n^{\varepsilon} $ and $ X\ll MN\ll X $.
Lemma 2.5. Let $ P^{\frac{8}{11}}\ll X\ll P $, $ H = X^{\frac{1}{58}} $ and $ c_h(\alpha) $ denote complex numbers such that $ c_h(\alpha)\ll(1+|h|)^{-1} $. Then uniformly with respect to $ \alpha\in(\tau, 1-\tau) $, we have
$ \begin{eqnarray} S_I = \sum\limits_{|h|\sim H}c_h(\alpha)\sum\limits_{M\leq m\leq2M}a(m)\sum\limits_{N\leq n\leq2N}e\left((h+\alpha)(mn)^c\right)\ll X^{\frac{57}{58}+2\varepsilon} \end{eqnarray} $ | (2.1) |
for any $ a(m)\ll m^\varepsilon $, where $ X\ll MN\ll X $ and $ M\ll X^{\frac{183}{290}} $.
Proof. We have
$ \begin{equation} S_I\ll X^\varepsilon\max\limits_{|\lambda|\in(\tau, H+1)}\sum\limits_{M\leq m\leq2M}\left|\sum\limits_{N\leq n\leq2N}e(\lambda(mn)^c)\right|. \end{equation} $ | (2.2) |
For the inner sum over $ n $ in (2.2), we have
$ \begin{equation*} \begin{split} S_I&\ll X^\varepsilon\max\limits_{|\lambda|\in(\tau, H+1)}\sum\limits_{M\leq m\leq2M}\left((|\lambda|X^c)^{\frac{1}{30}}N^{\frac{25}{30}}+\frac{N}{|\lambda|X^c}\right)\\ &\ll X^{\frac{57}{58}+2\varepsilon}, \end{split} \end{equation*} $ |
where Lemma 2.2 with the exponential pair $ (\kappa, \lambda) = (\frac{1}{30}, \frac{26}{30}) $ is used.
Lemma 2.6. Let $ P^{\frac{8}{11}}\ll X\ll P $, $ H = X^{\frac{1}{58}} $, and let $ c_h(\alpha) $ denote complex numbers such that $ c_h(\alpha)\ll(1+|h|)^{-1} $. Then uniformly with respect to $ \alpha\in(\tau, 1-\tau) $, we have
$ \begin{eqnarray} S_{II} = \sum\limits_{|h|\sim H}c_h(\alpha)\sum\limits_{M\leq m\leq2M}a(m)\sum\limits_{N\leq n\leq2N}b(n)e\left((h+\alpha){(mn)}^c\right)\ll X^{\frac{57}{58}+2\varepsilon} \end{eqnarray} $ | (2.3) |
for any $ a(m)\ll m^\varepsilon $, $ b(n)\ll n^\varepsilon $, $ X\ll MN\ll X $ and $ X^{\frac{1}{29}}\ll M\ll X^{\frac{83}{116}}. $
Proof. Taking $ Q = X^{\frac{1}{29}} $, then $ Q = o(N) $. According to Cauchy's inequality and Lemma 2.1, we get
$ \begin{eqnarray} |S_{II}| \ll X^{2\varepsilon}\sum\limits_{|h|\leq H}|c_h(\alpha)| \left(\frac{X^2}{Q}+\frac{X}{Q}\sum\limits_{q\leq Q}\sum\limits_{M\leq m\leq2M}\left|\sum\limits_{N\leq n\leq2N}e(f_h(m, n, q))\right|\right)^{\frac{1}{2}}, \end{eqnarray} $ | (2.4) |
where $ f_h(m, n, q) = (h+\alpha)n^c((m+q)^c-m^c) $. Thus, it is sufficient to estimate the sum
$ \begin{align*} S': = \sum\limits_{N\leq n\leq2N}e(f_h(m, n, q)). \end{align*} $ |
By Lemma 2.2 with the exponential pair $ (\kappa, \lambda) = (\frac{1}{6}, \frac{2}{3}) $, we have
$ \begin{align*} S'\ll\left((qHX^cM^{-1})^{\frac{1}{6}}N^{\frac{1}{2}}+\frac{X}{q\tau X^c}\right). \end{align*} $ |
Putting the above estimate into (2.4), we can obtain
$ \begin{eqnarray*} |S_{II}|&\ll&X^{2\varepsilon}\sum\limits_{|h|\leq H}|c_h(\alpha)|\left(\frac{X^2}{Q}+\frac{X}{Q}\sum\limits_{q\leq Q}\sum\limits_{M\leq m\leq2M}\left((qHX^cM^{-1})^{\frac{1}{6}}N^{\frac{1}{2}}+\frac{X}{q\tau X^c}\right)\right)^{\frac{1}{2}}\\ &\ll&X^{\frac{57}{58}+2\varepsilon}. \end{eqnarray*} $ |
Thus we complete the proof of Lemma 2.6.
Lemma 2.7. [16, Theorem 2] Suppose $ K > 1, \gamma > 0, c > 1, c\notin\mathbb{Z} $. Let $ \mathfrak{A}(K; c, \gamma) $ denote the number of solutions of the inequality
$ |n_1^c+n_2^c-n_3^c-n_4^c| < \gamma, \ K < n_1, n_2, n_3, n_4\leq2K, $ |
then we have
$ \mathfrak{A}(K;c, \gamma)\ll(\gamma K^{4-c}+K^2)K^\varepsilon. $ |
Lemma 2.8. For $ 1 < c < 3\ (c\neq2) $, we have
$ \begin{align*} \int_0^1|S(\alpha)|^4d\alpha\ll(P^{4-c}+P^2)P^\varepsilon. \end{align*} $ |
Proof. By a splitting argument, it is sufficient to show that
$ \begin{align*} \int_0^1\left|S\left(\alpha, \frac{P}{2}\right)\right|^4d\alpha\ll(P^{4-c}+P^2)P^\varepsilon. \end{align*} $ |
We have
$ \begin{eqnarray*} && \int_0^1\left|S\left(\alpha, \frac{P}{2}\right)\right|^4d\alpha\\ & = &\sum\limits_{\frac{P}{2} < p_1, p_2, p_3, p_4\leq P}(\log p_1)\cdots (\log p_4)\int_0^1e\left(([p_1^c]+[p_2^c]-[p_3^c]-[p_4^c])\alpha\right)d\alpha\\ & = &\sum\limits_{\frac{P}{2} < p_1, p_2, p_3, p_4\leq P\atop [p_1^c]+[p_2^c] = [p_3^c]+[p_4^c]}(\log p_1)\cdots (\log p_4)\ll(\log P)^4\sum\limits_{\frac{P}{2} < n_1, n_2, n_3, n_4\leq P\atop [n_1^c]+[n_2^c] = [n_3^c]+[n_4^c]}1. \end{eqnarray*} $ |
If $ [n_1^c]+[n_2^c] = [n_3^c]+[n_4^c] $, we can obtain
$ |n_1^c+n_2^c-n_3^c-n_4^c| = |\{n_1^c\}+\{n_2^c\}-\{n_3^c\}-\{n_4^c\}|\leq2. $ |
From Lemma 2.7, we have
$ \int_0^1|S(\alpha)|^4d\alpha\ll(\log P)^4\cdot\mathfrak{A}\left(\frac{P}{2};c, 2\right)\ll(P^{4-c}+P^2)P^\varepsilon, $ |
which completes the proof of Lemma 2.8.
Lemma 3.1. For $ \tau\leq\alpha\leq1-\tau $, we have
$ \begin{eqnarray*} S(\alpha)\ll P^{\frac{57}{58}+2\varepsilon}. \end{eqnarray*} $ |
Proof. Throughout the proof of this lemma, we write $ H = X^{\frac{1}{58}} $ for convenience. We need only to show that the estimation
$ \begin{equation} \sum\limits_{X < n\leq2X}\Lambda(n)e(\alpha[n^c])\ll X^{\frac{57}{58}+2\varepsilon} \end{equation} $ | (3.1) |
holds for $ P^{\frac{8}{11}}\leq X\leq P $ and $ \tau\leq\alpha\leq1-\tau $. By Lemma 2.3, we can obtain
$ \begin{eqnarray} \sum\limits_{X < n\leq2X}\Lambda(n)e(\alpha[n^c])& = &\sum\limits_{|h|\leq H}c_h(\alpha)\sum\limits_{X < n\leq2X}\Lambda(n)e((h+\alpha)n^c)\\ & &+O\left((\log X)\sum\limits_{X < n\leq2X}\min\left(1, \frac{1}{H\|n^c\|}\right)\right). \end{eqnarray} $ | (3.2) |
By the expansion
$ \begin{align} \min\left(1, \frac{1}{H\|n^c\|}\right) = \sum\limits_{h = -\infty}^\infty a_he(hn^c) \end{align} $ | (3.3) |
with
$ \begin{align} |a_h|\leq\min\left(\frac{\log2H}{H}, \frac{1}{|h|}, \frac{H}{h^2}\right), \end{align} $ | (3.4) |
we get
$ \begin{eqnarray} \sum\limits_{X < n\leq2X}\min\left(1, \frac{1}{H\|n^c\|}\right)&\leq& \sum\limits_{h = -\infty}^\infty a_h\left|\sum\limits_{X < n\leq2X}e(hn^c)\right|\\ &\ll& \frac{X\log2H}{H}+\sum\limits_{1\leq h\leq H}\frac{1}{h}\left((hX^c)^{\frac{1}{2}}+\frac{X}{hX^c}\right)\\ &&+ \sum\limits_{h > H}\frac{H}{h^2}\left((hX^c)^{\frac{1}{2}}+\frac{X}{hX^c}\right)\\ &\ll& X^{\frac{57}{58}}\log X, \end{eqnarray} $ | (3.5) |
where we estimate the sum over $ n $ by Lemma 2.2 with the exponent pair $ (\kappa, \lambda) = \left(\frac{1}{2}, \frac{1}{2}\right) $.
Taking $ U = X^{\frac{1}{29}} $, $ V = X^{\frac{1}{3}} $, $ Z = [X^{\frac{14}{29}}]+\frac{1}{2} $. By Lemma 2.4 with $ F(n) = e((h+\alpha)n^c) $, we get that the sum
$ \sum\limits_{|h|\leq H}c_h(\alpha)\sum\limits_{X < n\leq2X}\Lambda(n)e((h+\alpha)n^c) $ |
can be represented as $ O(\log^{10}X) $ sums, either of type Ⅰ
$ S_I' = \sum\limits_{|h|\leq H}c_h(\alpha)\sum\limits_{M < m\leq2M}a(m)\sum\limits_{N < n\leq2N}e((h+\alpha)(mn)^c), \ N > Z, $ |
or type Ⅱ
$ S_{II}' = \sum\limits_{|h|\leq H}c_h(\alpha)\sum\limits_{M < m\leq2M}a(m)\sum\limits_{N < n\leq2N}b(n)e((h+\alpha)(mn)^c), \ U < M < V. $ |
By Lemma 2.5, we get
$ \begin{equation} S_I'\ll X^{\frac{57}{58}+2\varepsilon}. \end{equation} $ | (3.6) |
By Lemma 2.6, we get
$ \begin{equation} S_{II}'\ll X^{\frac{57}{58}+2\varepsilon}. \end{equation} $ | (3.7) |
From (3.6) and (3.7), we can obtain
$ \begin{equation} \sum\limits_{|h|\leq H}c_h(\alpha)\sum\limits_{X < n\leq2X}\Lambda(n)e((h+\alpha)n^c)\ll X^{\frac{57}{58}+2\varepsilon}. \end{equation} $ | (3.8) |
From (3.2), (3.5), and (3.8), we complete the proof of Lemma 3.1.
Lemma 3.2. For $ \alpha\in(0, 1) $, we have
$ \begin{align*} T(\alpha, X)\ll X^{\frac{4+c}{7}}\log X+\frac{1}{\alpha X^{c-1}}. \end{align*} $ |
Proof. Let $ H' = X^{\frac{3-c}{7}} $. By Lemma 2.3, we obtain
$ \begin{eqnarray} \label{3.5} T(\alpha, X) = \sum\limits_{|h|\leq H'}c_h(\alpha)\sum\limits_{X < n\leq2X}e((h+\alpha)n^c) +O\left((\log X)\sum\limits_{X < n\leq2X}\min\left(1, \frac{1}{H'\parallel n^c\parallel}\right)\right). \end{eqnarray} $ |
From (3.3) and (3.4), we have
$ \begin{eqnarray} \sum\limits_{X < n\leq2X}\min\left(1, \frac{1}{H'\parallel n^c\parallel}\right)&\leq&\sum\limits_{h = -\infty}^\infty|a_h|\left|\sum\limits_{X < n\leq2X}e(hn^c)\right|\\ &\ll&\frac{X\log2H'}{H'}+\sum\limits_{1\leq h\leq H'}\frac{1}{h}\left((hX^c)^{\frac{1}{6}}X^{\frac{1}{2}}+\frac{X}{hX^c}\right)\\ &&+\sum\limits_{h\geq H'}\frac{H'}{h^2}\left((hX^c)^{\frac{1}{6}}X^{\frac{1}{2}}+\frac{X}{hX^c}\right)\\ &\ll&X^{\frac{c+4}{7}}\log X, \end{eqnarray} $ | (3.9) |
where we used Lemma 2.2 with the exponent pair $ (\kappa, \lambda) = (\frac{1}{6}, \frac{2}{3}) $. Similarly, we have
$ \begin{eqnarray} &&\sum\limits_{|h|\leq H}c_h(\alpha)\sum\limits_{X < n\leq2X}e((h+\alpha)n^c)\\ & = &c_0(\alpha)\sum\limits_{X < n\leq2X}e(\alpha n^c)+\sum\limits_{1\leq|h|\leq H'}c_h(\alpha)\sum\limits_{X < n\leq2X}e((h+\alpha)n^c)\\ &\ll&X^{\frac{c+4}{7}}\log X+\frac{X}{\alpha X^c}. \end{eqnarray} $ | (3.10) |
From (3.9) and (3.10), we complete the proof of Lemma 3.2.
Lemma 3.3. Let $ P^{\frac{8}{11}}\ll X\ll P $, we have
$ \begin{align} \int_{\tau}^{1-\tau}|S(\alpha)|^{4} d \alpha \ll P^{\frac{431-87c}{116}+\varepsilon}+P^{\frac{2734-377c}{812}+ \varepsilon}+P^{\frac{36+2 c}{14}+\varepsilon}. \end{align} $ | (3.11) |
Proof. Let $ \Theta = (\tau, 1-\tau) $ and $ K_l(\alpha) = \overline{S(\alpha)}|S(\alpha)|^l $ ($ l = 1 $ or 2). Then we have
$ \begin{eqnarray} &&\left|\int_\Theta S(\alpha)K_l(\alpha)d\alpha\right|\\ &\ll&(\log N) \max _{P^{\frac{8}{11}} \leq X \leq P}\left|\int_{\Theta} S(\alpha, X) K_{l}(\alpha) d \alpha\right|+P^{\frac{8}{11}}(\log P) \int_{\Theta}\left|K_{l}(\alpha)\right| d \alpha. \end{eqnarray} $ | (3.12) |
In addition, we get
$ \begin{eqnarray} \left|\int_{\Theta} S(\alpha, X) K_{l}(\alpha) d \alpha\right|& = &\left|\sum\limits_{X \leq p \leq 2 X} \log p \int_{\Theta} e\left(\alpha\left[p^{c}\right]\right) K_{l}(\alpha) d \alpha\right| \\ &\leq& \sum\limits_{X \leq p \leq 2 X} \log p\left|\int_{\Theta} e\left(\alpha\left[p^{c}\right]\right) K_{l}(\alpha) d \alpha\right|\\ &\leq&(\log X) \sum\limits_{X \leq n \leq 2 X}\left|\int_{\Theta} e\left(\alpha\left[n^{c}\right]\right) K_{l}(\alpha) d \alpha\right|. \end{eqnarray} $ | (3.13) |
By (3.13) and Cauchy's inequality, we have
$ \begin{eqnarray} &&\left|\int_{\Theta} S(\alpha, X) K_{l}(\alpha) d \alpha\right|^{2} \leq X\left(\log ^{2} X\right) \sum\limits_{X \leq n \leq 2 X}\left|\int_{\Theta} e\left(\alpha\left[n^{c}\right]\right) K_{l}(\alpha) d \alpha\right|^{2} \\ &\leq& X\left(\log ^{2} X\right) \int_{\Theta} \overline{K_{l}(\beta)} d \beta \int_{\Theta} K_{l}(\alpha) T(\alpha-\beta, X) d \alpha \\ &\leq& X\left(\log ^{2} X\right) \int_{\Theta}\left|K_{l}(\beta)\right| d \beta \int_{\Theta}\left|K_{l}(\alpha) \| T(\alpha-\beta, X)\right| d \alpha. \end{eqnarray} $ | (3.14) |
Then,
$ \begin{eqnarray} & & \int_{\Theta}\left|K_{l}(\alpha) T(\alpha-\beta, X)\right| d \alpha \\ &\ll& \int_{\Theta \atop|\alpha-\beta| \leq X^{-c}}\left|K_{l}(\alpha) T(\alpha-\beta, X)\right| d \alpha+\int_{\Theta\atop{|\alpha-\beta| > X^{-c}}}\left|K_{l}(\alpha) T(\alpha-\beta, X)\right| d \alpha. \end{eqnarray} $ | (3.15) |
For the first integral in (3.15), we use the trivial bound $ T(\alpha, X) \leq X $ and get
$ \begin{eqnarray} & &\int_{\Theta\atop{|\alpha-\beta| \leq X^{-c}}}\left|K_{l}(\alpha) T(\alpha-\beta, X)\right| d \alpha \\ &\ll& X \max _{\alpha \in \Theta}\left|K_{l}(\alpha)\right| \int_{|\alpha-\beta| \leq X^{-c}} 1 d \alpha \ll X^{1-c} \max _{\alpha \in \Theta}\left|K_{l}(\alpha)\right|. \end{eqnarray} $ | (3.16) |
For the second integral in (3.15), we use Lemma 3.2 and get
$ \begin{eqnarray} &&\int_{\Theta \atop{|\alpha-\beta| > X^{-c}}}\left|K_{l}(\alpha) \| T(\alpha-\beta, X)\right| d \alpha\\ &\ll& \int_{\Theta \atop|\alpha-\beta| > X^{-c}}\left|K_{l}(\alpha)\right|\left(X^{\frac{4+ c}{7}} \log X+\frac{X^{1-c}}{|\alpha-\beta|}\right) d \alpha\\ &\ll& X^{\frac{4+ c}{7}} \log X \int_{\Theta}\left|K_{l}(\alpha)\right| d \alpha+X^{1-c} \max _{\alpha \in \Theta}\left|K_{l}(\alpha)\right| \int_{X-c < |\alpha-\beta| \leq 2} \frac{1}{|\alpha-\beta|} d \alpha\\ &\ll& X^{\frac{4+c}{7}}(\log X) \int_{\Theta}\left|K_{l}(\alpha)\right| d \alpha+X^{1-c} \max _{\alpha \in \Theta}\left|K_{l}(\alpha)\right| \log X. \end{eqnarray} $ | (3.17) |
From (3.12) and (3.14)–(3.17), we obtain
$ \begin{eqnarray} \left|\int_{\Theta} S(\alpha) K_{l}(\alpha) d \alpha\right|^{2} &\ll& X^{\frac{11+c}{7}+\varepsilon}\left(\int_{\Theta}\left|K_{l}(\alpha)\right| d \alpha\right)^{2} +P^{\frac{16}{11}+\varepsilon}\left(\int_{\Theta}\left|K_{l}(\alpha)\right| d \alpha\right)^{2}\\ &&+X^{2-c+\varepsilon} \max _{\alpha \in \Theta}\left|K_{l}(\alpha)\right| \int_{\Theta}\left|K_{l}(\alpha)\right| d \alpha. \end{eqnarray} $ | (3.18) |
By applying Lemma 3.1 and the bound
$ \begin{align*} \int_{\tau}^{1-\tau}|S^2(\alpha)|d\alpha\ll\int_0^1|S^2(\alpha)|d\alpha\ll P\log^2P, \end{align*} $ |
we can deduce from (3.18) with $ l = 1 $ that
$ \begin{eqnarray} && \int_{\Theta}|S(\alpha)|^{3} d \alpha = \int_{\Theta} S(\alpha) K_{1}(\alpha) d \alpha \\ &\ll& X^{1-\frac{c}{2}+\varepsilon} \max _{\alpha \in \Theta}|S(\alpha)|\left(\int_{0}^{1}|S(\alpha)|^{2} d \alpha\right)^{\frac{1}{2}} \\ &&+\left(X^{\frac{11+ c}{14}+\varepsilon}+P^{\frac{8}{11}+\varepsilon}\right)\left(\int_{0}^{1}|S(\alpha)|^{2} d \alpha\right) \\ &\ll& X^{1-\frac{c}{2}} P^{\frac{57}{58}+\frac{1}{2}+ \varepsilon}+X^{\frac{11+ c}{14}} P^{1+ \varepsilon}+P^{\frac{19}{11}+\varepsilon} \\ &\ll& P^{\frac{144-29c}{58}+ \varepsilon}+P^{\frac{25+ c}{14}+\varepsilon}. \end{eqnarray} $ | (3.19) |
Then it follows from (3.18) with $ r = 2 $ and (3.19) that
$ \begin{eqnarray*} &&\int_{\Theta}|S(\alpha)|^{4} d \alpha = \int_{\Theta} S(\alpha) K_{2}(\alpha) d \alpha\nonumber\\ &\ll& X^{1-\frac{c}{2}+\varepsilon} \max _{\alpha \in \Theta}|S(\alpha)|^{\frac{3}{2}}\left(\int_{\Theta}|S(\alpha)|^{3} d \alpha\right)^{\frac{1}{2}} +\left(X^{\frac{11+ c}{14}+\varepsilon}+P^{\frac{8}{11}+\varepsilon}\right)\left(\int_{\Theta}|S(\alpha)|^{3} d \alpha\right)\nonumber\\ &\ll& P^{\frac{287}{116}-\frac{c}{2}+ \varepsilon}\left(P^{\frac{144-29c}{58}}+P^{\frac{25+ c}{14}}\right)^{\frac{1}{2}}+P^{\frac{11+c}{14}+ \varepsilon}\left(P^{\frac{144-29c}{58}}+P^{\frac{25+ c}{14}}\right)\nonumber\\ &\ll& P^{\frac{431-87c}{116}+\varepsilon}+P^{\frac{1327-174c}{406}+ \varepsilon}+P^{\frac{36+2 c}{14}+\varepsilon}, \end{eqnarray*} $ |
which completes the proof of Lemma 3.3.
By the definition of $ R(n) $, we have
$ \begin{eqnarray} R(n)& = &\int_{-\tau}^{1-\tau}S^4(\alpha)e(-\alpha n)d\alpha \\ & = &\int_{-\tau}^{\tau}S^4(\alpha)e(-\alpha n)d\alpha+\int_{\tau}^{1-\tau}S^4(\alpha)e(-\alpha n)d\alpha\\ & = &R_1(N)+R_2(N). \end{eqnarray} $ | (4.1) |
In this subsection, we shall prove the following equation
$ \begin{align} R_1(n) = \frac{\Gamma^4(1+\frac{1}{c})}{\Gamma(\frac{4}{c})}n^{\frac{4}{c}-1}+O\left(N^{\frac{4}{c}-1}\exp(-(\log n)^{\frac{1}{4}})\right). \end{align} $ | (4.2) |
Define
$ \begin{eqnarray*} &&G(\alpha) = \sum\limits_{m\leq N}\frac{1}{c}m^{\frac{1}{c}-1}e(m\alpha), \\ &&B_1(n) = \int_{-\tau}^\tau G^4(\alpha)e(-n\alpha)d\alpha, \\ &&B(n) = \int_{-\frac{1}{2}}^{\frac{1}{2}}G^4(\alpha)e(-n\alpha)d\alpha. \end{eqnarray*} $ |
Then
$ \begin{align} R_1(n) = (R_1(n)-B_1(n))+(B_1(n)-B(n))+B(n). \end{align} $ | (4.3) |
As is shown in Theorem 2.3 of Vaughan [17], we can obtain
$ \begin{align} B(n) = \frac{\Gamma^4(1+\frac{1}{c})}{\Gamma(\frac{4}{c})}P^{4-c}+O(P^{3-c}). \end{align} $ | (4.4) |
From Lemma 2.8 of Vaughan [17], for $ \nu > 0 $, we have
$ \begin{align} B_1(n)-B(n)\ll\int_\tau^{\frac{1}{2}}|G(\alpha)|^4d\alpha\ll\int_\tau^{\frac{1}{2}} \alpha^{-\frac{4}{c}}d\alpha\ll\tau^{1-\frac{4}{c}}\ll P^{4-c-\nu}. \end{align} $ | (4.5) |
Next we estimate $ |R_1(n)-B_1(n)| $. Let $ W_1(N) $ denote the set of integers $ n $ in the interval $ (\frac{N}{2}, N] $ such that
$ \begin{align} |R_1(n)-B_1(n)| = \left|\int_{-\tau}^{\tau}(S^4(\alpha)-G^4(\alpha))e(-n\alpha)d\alpha\right|\geq\frac{n^{\frac{4}{c}-1}}{\log n}. \end{align} $ | (4.6) |
We take $ W_1 = |W_1(N)| $, and choose the complex number $ \varphi_1(n) $ satisfying $ |\varphi_1(n)| = 1 $ and
$ \begin{align} \varphi_1(n)\int_{-\tau}^{\tau}(S^4(\alpha)-G^4(\alpha))e(-n\alpha)d\alpha = \left|\int_{-\tau}^{\tau}(S^4(\alpha)-G^4(\alpha))e(-n\alpha)d\alpha\right|. \end{align} $ | (4.7) |
Thus, for $ n\in W_1(N) $, by (4.6) and (4.7), we can obtain
$ \begin{align} \frac{N^{\frac{4}{c}-1}W_1}{\log N}\ll\int_{-\tau}^{\tau}(S^4(\alpha)-G^4(\alpha))L(\alpha)d\alpha, \end{align} $ | (4.8) |
where $ L(\alpha) = \sum_{n\in W_1(N)}\varphi_1(n)e(-\alpha n) $. By Lemma 2.8 of Vaughan [17], we have
$ \begin{align*} G(\alpha)\ll\min(N^{\frac{1}{c}}, |\alpha|^{-\frac{1}{c}}). \end{align*} $ |
Thus,
$ \begin{eqnarray} \int_{-\frac{1}{2}}^{\frac{1}{2}}|G(\alpha)|^4d\alpha&\ll&\int_0^{\frac{1}{2}}\min(N^{\frac{1}{c}}, |\alpha|^{-\frac{1}{c}})^4d\alpha\\ &\ll&\int_0^{\frac{1}{N}}N^{\frac{4}{c}}d\alpha+\int_{\frac{1}{N}}^{\frac{1}{2}}\alpha^{-\frac{4}{c}}d\alpha\\ &\ll&N^{\frac{4}{c}-1}\ll P^{4-c}. \end{eqnarray} $ | (4.9) |
For $ |\alpha|\leq\tau $, we have
$ \begin{align} S(\alpha) = \sum\limits_{p\leq P}(\log p)e(p^c\alpha)+O(\tau P) = S^*(\alpha)+O(\tau P). \end{align} $ | (4.10) |
Now we consider the upper bound of $ |S(\alpha)-G(\alpha)| $ under the condition $ |\alpha|\leq\tau $. By (4.10), we have
$ \begin{eqnarray} S(\alpha)& = &\sum\limits_{n\leq P}\Lambda(n)e(n^c\alpha)+O(P^{\frac{1}{2}})+O(\tau P)\\ & = &\sum\limits_{n\leq P}\Lambda(n)e(n^c\alpha)+O(P^{1-\varepsilon}). \end{eqnarray} $ | (4.11) |
For $ |\alpha|\leq\tau $ and $ u\geq2 $, by Lemma 1.2 of Ivić [18], we have
$ \begin{align*} \sum\limits_{1 < m\leq u}e(m\alpha) = \int_1^ue(t\alpha)dt+O(1). \end{align*} $ |
According to partial summation and the above identity, we can obtain
$ \begin{eqnarray} \sum\limits_{n\leq P}\Lambda(n)e(n^c\alpha)& = &\int_1^Pe(t^c\alpha)dt+O(P\exp(-(\log P)^{\frac{1}{3}}))\\ & = &\int_1^N\frac{1}{c}u^{\frac{1}{c}-1}e(u\alpha)du+O(P\exp(-(\log P)^{\frac{1}{3}}))\\ & = &\sum\limits_{m\leq N}\frac{1}{c}m^{\frac{1}{c}-1}e(m\alpha)+O(P\exp(-(\log P)^{\frac{1}{3}}))\\ & = &G(\alpha)+O(P\exp(-(\log P)^{\frac{1}{3}})). \end{eqnarray} $ | (4.12) |
By (4.11) and (4.12), we have
$ \begin{align} \sup\limits_{|\alpha|\leq\tau}|S(\alpha)-G(\alpha)|\ll P\exp(-(\log P)^{\frac{1}{3}}). \end{align} $ | (4.13) |
We use Hölder's inequality, Lemma 2.8, (4.9), (4.13), and the obvious bound $ \int_0^1|L(\alpha)|^4d\alpha\ll W_1^3 $ and get
$ \begin{eqnarray} &&\int_{-\tau}^{\tau}(S^4(\alpha)-G^4(\alpha))L(\alpha)d\alpha\\ &\ll&\int_{-\tau}^\tau|S(\alpha)-G(\alpha)|(|S(\alpha)|^3+|G(\alpha)|^3)|L(\alpha)|d\alpha\\ &\ll&\sup\limits_{|\alpha|\leq\tau}|S(\alpha)-G(\alpha)|\left(\int_{-\tau}^\tau|S(\alpha)|^4+|G(\alpha)|^4d\alpha\right)^{\frac{3}{4}} \left(\int_{-\tau}^{1-\tau}|L(\alpha)|^4d\alpha\right)^{\frac{1}{4}}\\ &\ll& N^{\frac{4}{c}-\frac{3}{4}}\exp(-\log^{\frac{1}{4}}N)W_1^{\frac{3}{4}}. \end{eqnarray} $ | (4.14) |
From (4.8) and (4.14), we obtain
$ \begin{align*} \frac{N^{\frac{4}{c}-1}W_1}{\log N}\ll N^{\frac{4}{c}-\frac{3}{4}}\exp(-\log^{\frac{1}{4}}N)W_1^{\frac{3}{4}}, \end{align*} $ |
which yields that
$ \begin{align} W_1\ll N\exp(-\log^{\frac{1}{5}}N). \end{align} $ | (4.15) |
In this subsection, let $ W_2(N) $ denote the set of integers $ n $ in the interval $ (\frac{N}{2}, N] $ such that
$ \begin{align} |R_2(n)| = \left|\int_\tau^{1-\tau}S^4(\alpha)e(-n\alpha)d\alpha\right|\gg \frac{n^{\frac{4}{c}-1}}{\log n}. \end{align} $ | (4.16) |
By Bessel's inequality and taking $ W_2 = |W_2(N)| $, we have
$ \begin{align} W_2\left(\frac{N^{\frac{4}{c}-1}}{\log N}\right)^2\ll\sum\limits_{n\in W_2(N)}\left|\int_\tau^{1-\tau}S^4(\alpha)e(-n\alpha)d\alpha\right|^2\ll\int_\tau^{1-\tau}|S^{8}(\alpha)|d\alpha. \end{align} $ | (4.17) |
Since $ 1 < c < \frac{38}{29} $ and $ \varepsilon\in(0, 10^{-10}(\frac{38}{29}-c)) $, we can deduce from (4.17) and Lemma 3.3 that
$ \begin{eqnarray} W_2&\ll& N^{2-\frac{8}{c}+\varepsilon}P^{\frac{57}{58}\times4+\varepsilon}\left(P^{\frac{431-87c}{116}+\varepsilon}+P^{\frac{1327-174c}{406}+ \varepsilon}+P^{\frac{36+2 c}{14}+\varepsilon}\right)\\ &\ll& P^{c-\varepsilon}\ll N^{1-\varepsilon}. \end{eqnarray} $ | (4.18) |
Let $ \mathcal{W}(N) $ denote the number of integers $ n $ in the interval $ (\frac{N}{2}, N] $ such that
$ \begin{align} \left|R(n)-\frac{\Gamma^3\left(1+\frac{1}{c}\right)}{\Gamma\left(\frac{4}{c}\right)}n^{\frac{4}{c}-1}\right|\geq\frac{n^{\frac{4}{c}-1}}{\log n}. \end{align} $ | (4.19) |
Then, by (4.1)–(4.4), (4.15), and (4.18), we have
$ \begin{equation*} \left|R(n)-\frac{\Gamma^3\left(1+\frac{1}{c}\right)}{\Gamma\left(\frac{3}{c}\right)}n^{\frac{4}{c}-1}\right|\leq|R_1(n)-B_1(n)|+|R_2(n)|+O\left(n^{\frac{4}{c}-1-\varepsilon}\right). \end{equation*} $ |
From the above formula and (4.19), we can get
$ \begin{align*} \mathcal{W}(N)\leq W_1+W_2\ll N \exp(-\log^{\frac{1}{5}}N). \end{align*} $ |
Thus we complete the proof of Theorem 1.
In this paper, we proved that almost all $ n\in(N, 2N] $ can be represented as $ n = [p_1^c]+[p_2^c]+[p_3^c]+[p_4^c] $, where $ p_1, p_2, p_3, p_4 $ are prime numbers and $ [x] $ denotes the integer part of $ x $. Our method also yields an asymptotic formula for the number of representations of these $ n $.
J. Huang: Conceptualization, formal analysis, investigation, resources, writing—original draft, writing—review and editing; W. G. Zhai: Conceptualization, investigation, writing—original draft, writing—review and editing; D. Y. Zhang: Data curation, funding acquisition, methodology, project administration, supervision, validation, visualization, writing—review and editing. All authors have read and agreed to the published version of the manuscript.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work is supported by National Natural Science Foundation of China (Grant Nos. 12171286, 12471009) and Beijing Natural Science Foundation (Grant No. 1242003).
The authors declare there is no conflict of interest.
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