1 Introduction

The famous Goldbach conjecture states that any even number greater than 2 can be written as the sum of two prime numbers. Let \(P_{r}\) denote an almost-prime with at most r prime factors counted with multiplicity. In 1966, Chen [2] announced his remarkable theorem–Chen’s theorem: every sufficiently large even integer N can written as

$$\begin{aligned} N=p+P_{2}, \end{aligned}$$

where and in what follows p, with or without subscript, is a prime. And the detail was published in [3].

The ancient Waring problem says that for every natural number \(k\ge 2\) there exists a positive integer \(s=s(k)\) such that every natural number is a sum of at most s kth powers of natural numbers. In 1934, Segal [12] generalized the Waring problem to fractional exponents. And, he showed that for any fixed real number \(c>1,\) there exists a positive integer \(s=s(c)\) such that every sufficiently large natural number N can be written as

$$\begin{aligned} N=[x_{1}^{c}]+[x_{2}^{c}]+\cdots +[x_{s}^{c}], \end{aligned}$$

where \(x_{1},x_{2},\dots ,x_{s}\) are non-negative integers. On the other hand, some mathematicians consider that how large c can be for the fixed \(s\ge 2\). In 1973, Deshouillers [4] proved that if \(1<c<4/3,\) then every sufficiently large natural number N can be represented as

$$\begin{aligned} N=[x_{1}^{c}]+[x_{2}^{c}], \end{aligned}$$

where \(x_{1}\) and \(x_{2}\) are non-negative integers. Later, the domain of c was improved to \(1<c<55/41\) and \(1<c<3/2,\) respectively by Gritsenko [7] and Konyagin [9]. In addition, in 2009, Kumchev [10] proved that if \(1<c<16/15,\) then every sufficiently large natural number N can be represented as

$$\begin{aligned} N=[p^{c}]+[m^{c}], \end{aligned}$$
(1.1)

where m is a positive integer. Recently, the range of c obtained by Kumchev was improved by Yu [14] to \(1<c<11/10.\) Furthermore, Petrov and Tolev [11] proved that if \(1<c<29/28,\) then every sufficiently large natural number N can be represented as (1.1) with m is an almost prime with at most \([52/(29-28c)]+1\) prime factors. Inspired by Chen’s theorem, Petrov and Tolev [11] proposed the following interesting conjecture:

Conjecture 1.1

There exists a constant \(c_{0}>1\) such that if \(1<c<c_{0}\), then every sufficiently large natural number N can be represented as

$$\begin{aligned} N=[p^{c}]+[P_{2}^{c}]. \end{aligned}$$

In the present paper, we improve the results of Petrov and Tolev when c close to 1. And, we state our theorem in the following.

Theorem 1.1

Suppose that \(1<c\le 1.0198.\) Then every sufficiently large natural number N can be represented as

$$\begin{aligned} N=[p^{c}]+[m^{c}], \end{aligned}$$

where p is a prime and m is an almost prime with at most 10 prime factors.

Remark 1.1

While our method does not yield a general formula for the number of prime factors of m in terms of c,  for every specific \(c\in (1, 29/28),\) one can apply our method to get an improvement to Petrov and Tolev’s result.

2 Notation and Preliminaries

From now on, let N be a sufficiently large natural number and

$$\begin{aligned} 1<c\le 1.0198, \ \ \theta =\frac{1}{c} \end{aligned}$$

be the positive real numbers. Put

$$\begin{aligned} P=\delta N^{\theta },\ \ \delta =10^{-9}. \end{aligned}$$

Put \(e(y)= e^{2\pi iy}.\) As usual, \(\mu (n)\) denotes the Möbius function. Let \(\rho (t)=\frac{1}{2}-\{t\}\), where \(\{t\}\) is the fractional part of t. Define \((\xi _{d}^{+})\) and \((\xi _{d}^{-})\) the upper bound and lower bound beta-sieves of level D respectively (see Chapter 11 of [5]), for which we have

$$\begin{aligned} \sum _{d|n}\xi ^{-}(d)\le \sum _{d|n}\mu (d)\le \sum _{d|n}\xi ^{+}(d). \end{aligned}$$
(2.1)

For \(z\ge 2,\) define

$$\begin{aligned} P(z)=\prod _{p<z}p\ \ \ \ \text{ and }\ \ \ \ V(z)=\prod _{p<z}\left( 1-\frac{1}{p}\right) . \end{aligned}$$

Then, by Theorem 11.12 of Friedlander and Iwaniec [5], we have

$$\begin{aligned} \sum _{d|P(z)}\frac{\xi ^{+}(d)}{d}\le V(z)\left( F(s)+o(1)\right) \end{aligned}$$
(2.2)

and

$$\begin{aligned} \sum _{d|P(z)}\frac{\xi ^{-}(d)}{d}\ge V(z)\left( f(s)+o(1)\right) , \end{aligned}$$
(2.3)

where F(s) and f(s) are the standard upper and lower bound functions of the linear sieve, and

$$\begin{aligned} s=\frac{\log D}{\log z}. \end{aligned}$$

Lemma 2.1

For F(s) and f(s),  We have

$$\begin{aligned} F(s)&=\frac{2e^{\gamma }}{s},\ \ 0<s\le 3;\\ F(s)&=\frac{2e^{\gamma }}{s}\Bigg (1+\int _{2}^{s-1}\frac{\log (t-1)}{t}\Bigg ),\ \ 3\le s\le 5;\\ F(s)&=\frac{2e^{\gamma }}{s}\Bigg (1+\int _{2}^{s-1}\frac{\log (t-1)}{t}+\int _{2}^{s-3}\frac{\log (t-1)}{t}dt\\&\quad \int _{t+2}^{s-1}\frac{1}{u}\log \frac{u-1}{t+1}\Bigg ),\ \ 5\le s\le 7;\\ f(s)&=0,\ \ 0< s\le 2;\\ f(s)&=\frac{2e^{\gamma }\log (s-1)}{s},\ \ 2\le s\le 4;\\ f(s)&=\frac{2e^{\gamma }}{s}\Bigg (\log (s-1)+\int _{3}^{s-1}\frac{dt}{t}\int _{2}^{t-1}\frac{\log (u-1)}{u}du\Bigg ),\ \ 4\le s\le 6;\\ f(s)&=\frac{2e^{\gamma }}{s}\Bigg (\log (s-1)+\int _{3}^{s-1}\frac{dt}{t}\int _{2}^{t-1}\frac{\log (u-1)}{u}du\\&\quad +\int _{2}^{s-4}\frac{\log (t-1)}{t}dt\int _{t+2}^{s-2}\frac{1}{u}\log \frac{u-1}{t+1}\log \frac{s}{u+2}du\Bigg ),\ \ 6\le s\le 8. \end{aligned}$$

Proof

See [8, (3.11) and (3.12)]. \(\square \)

We denote

$$\begin{aligned} D=N^{\eta }, \end{aligned}$$

where \(\eta =\eta (c)>0\) is a constant. Let

$$\begin{aligned} \Sigma _{j}=\sum _{d\mid P(z)}\xi _{d}^{-}\sum _{P<p\le 2P}(\log p) \rho \Big (-\frac{1}{d}(N+j-[p^{c}])^{\theta }\Big ),\ \ j=0,1. \end{aligned}$$
(2.4)

Lemma 2.2

Let

$$\begin{aligned} \frac{28}{29}<\theta<1,\ \ \eta <\frac{29\theta -28}{26}. \end{aligned}$$

Then we have

$$\begin{aligned} \Sigma _{0}, \Sigma _{1}\ll \frac{N^{2\theta -1}}{(\log N)^{2}},\ \ j=0,1. \end{aligned}$$

Proof

See (23),  (24),  (71),  and (73) of [11]. \(\square \)

Lemma 2.3

(Vaaler’s theorem) For each \(H\ge 2\) there are numbers \(c_{h}\), \(1\le h\le H,\) and \(d_{h},\) \(0\le h\le H,\) such that

$$\begin{aligned} \rho (t)=\sum _{1\le |h|\le H}c_{h}e(ht)+\Delta _{H}(t), \end{aligned}$$

where

$$\begin{aligned} |\Delta _{H}(t)|\le \sum _{0\le |h|\le H}d_{h}e(ht) \end{aligned}$$

and

$$\begin{aligned} |c_{h}|\ll \frac{1}{|h|},\ \ \ \ |d_{h}|\ll \frac{1}{H}. \end{aligned}$$

Proof

See [13].

Lemma 2.4

(Van der Corput’s Theorem) Suppose that \(\vartheta \) is a real valued function with two continuous derivatives on interval I. Suppose also that there is some \(\lambda >0\) such that

$$\begin{aligned} |\vartheta ^{\prime \prime }|\asymp \lambda \end{aligned}$$

on I. Then

$$\begin{aligned} \sum _{n\in I}e(\vartheta (n))\ll |I|\lambda ^{1/2}+\lambda ^{-1/2}. \end{aligned}$$

Proof

See [6, Theorem 2.2]. \(\square \)

3 A Key Mean Estimation

In this section, we prove a mean estimation similar to Lemma 2.2, which plays a crucial role in the proof of Theorem 1.1.

From now on, we take \(z=N^{\frac{1}{200}}.\) Let

$$\begin{aligned} R^{(k)}_{j}=\sum _{d\mid P(z)}\xi _{d}^{+}\sum _{m\in \mathcal {M}_{k}}\rho \Big (-\frac{1}{d}(N+j-[m^{c}])^{\theta }\Big ),\ \ j=0,1, \end{aligned}$$
(3.1)

where

$$\begin{aligned} \mathcal {M}_{k}&=\{m=p_{1}\cdots p_{k}:(1-(2\delta )^{c})^{\theta }N^{\theta }-1\le p_{1}\cdots p_{k}\nonumber \\&<(1-\delta ^{c})^{\theta }N^{\theta }+1, z\le p_{1}\le \dots \le p_{k}\}. \end{aligned}$$
(3.2)

By the prime number theorem, we have

$$\begin{aligned} |\mathcal {M}_{k}|&\le (1+o(1))\sum _{z\le p_{1}\le \dots \le p_{k-1}\le \Big (\frac{(1-\delta ^{c})^{\theta }N^{\theta }+1}{p_{1}\dots p_{k-2}}\Big )^{\frac{1}{2}}}\\&\quad \Big (\frac{(1-\delta ^{c})^{\theta }N^{\theta }+1}{p_{1}\dots p_{k-1}\log \frac{(1-\delta ^{c})^{\theta }N^{\theta }+1}{p_{1}\dots p_{k-1}}}-\frac{(1-(2\delta )^{c})^{\theta }N^{\theta }-1}{p_{1}\dots p_{k-1}\log \frac{(1-(2\delta )^{c})^{\theta }N^{\theta }-1}{p_{1}\dots p_{k-1}}}\Big ) \end{aligned}$$

Taking \(x=(1-\delta ^{c})^{\theta }N^{\theta }+1\) in [1, (4.29)] and by some routine arguments we get that

$$\begin{aligned} |\mathcal {M}_{k}|\le&((1-\delta ^{c})^{\theta }-(1-(2\delta )^{c})^{\theta }+o(1))c_{k}\frac{N^{\theta }}{\log N^{\theta }}, \end{aligned}$$
(3.3)

where

$$\begin{aligned} c_{k}=\int _{k-1}^{199}\frac{dt_{1}}{t_{1}}\int _{k-2}^{t_{1}-1}\frac{dt_{2}}{t_{2}}\cdots \int _{3}^{t_{k-4}-1}\frac{dt_{k-3}}{t_{k-3}}\int _{2}^{t_{k-3}-1}\frac{\log (t_{k-2}-1)dt_{k-2}}{t_{k-2}}. \end{aligned}$$

To compute the bound \(c_{k}\) we used the Mathematica technical computing software. For example, we use the following code to calculate \(c_{11}.\)

NIntegrate [(Log[t9 − 1]/t9)*(1/t8)*(1/t7)*(1/t6)*(1/t5)*(1/t4)*(1/t3)*(1/t2)*(1/t1), {t1, 10, 199}, {t2, 9, t1 − 1}, {t3, 8, t2 − 1}, {t4, 7, t3 − 1}, {t5, 6, t4 − 1}, {t6, 5, t5 − 1}, {t7, 4, t6 − 1}, {t8, 3, t7 − 1}, {t9, 2, t8 − 1}]

In fact, the estimate of \(c_{k}\) for \(k\ge 15\) has already been given in [1, (4.30)]. Whatever, we have

$$\begin{aligned}&c_{11}<0.580195,\ \ c_{12}<0.185152,\ \ c_{13}<0.052602,\ \ c_{14}<0.018655,\\&c_{15}<0.003088,\ \ c_{16}<0.000646,\ \ c_{17}<0.000124,\ \ c_{18}<0.000011,\\&c_{k}<0.000001 \ \ \text{ for }\ \ 19\le k\le 199. \end{aligned}$$

Hence, we have

$$\begin{aligned} \sum _{k=11}^{199}|\mathcal {M}_{k}|\le (0.840654+o(1))((1-\delta ^{c})^{\theta }-(1-(2\delta )^{c})^{\theta })\frac{N^{\theta }}{\log N^{\theta }}. \end{aligned}$$
(3.4)

Proposition 3.1

Let

$$\begin{aligned} \frac{28}{29}<\theta<1,\ \ \eta <\frac{29\theta -28}{26}. \end{aligned}$$

If \( k\ge 2,\) then we have

$$\begin{aligned} R_{0}^{(k)}+R_{1}^{(k)}\ll \frac{N^{2\theta -1}}{(\log N)^{3}}. \end{aligned}$$

Proof

Let \(Z\ge 2\) be any fixed integer. From page S43 and S44 of [11], we know that there exists a series of periodic functions \(g_{s}(t), s=0,1,\dots ,2Z-1\) with a period of 1 and has the following properties:

$$\begin{aligned}&0<g_{s}(t)\le 1\ \ \text{ for }\ \ \left| t-\frac{s}{2Z}\right| <\frac{1}{2Z}, \end{aligned}$$
(3.5)
$$\begin{aligned}&g_{s}(t)=0\ \ \text{ for }\ \ \frac{1}{2Z}<\left| t-\frac{s}{2Z}\right| <\frac{1}{2}, \end{aligned}$$
(3.6)

and

$$\begin{aligned} \sum _{s=0}^{2Z-1}g_{s}(t)=1\ \ \text{ for } \text{ all }\ \ t\in {{\mathbb {R}}}. \end{aligned}$$
(3.7)

Furthermore, we have

$$\begin{aligned} g_{s}(t)=\sum _{|n|\le Z(\log N)^{4}}\beta ^{(s)}_{n}e(nt)+O(N^{-\log \log N}),\ \ s=0,1,\dots ,2Z-1, \end{aligned}$$
(3.8)

where

$$\begin{aligned} \beta ^{(s)}_{n}\le \frac{1}{2Z}. \end{aligned}$$
(3.9)

By Lemma 2.3, we can write

$$\begin{aligned} R_{j}^{(k)}=R_{j1}^{(k)}+R_{j2}^{(k)}, \end{aligned}$$
(3.10)

where

$$\begin{aligned} R_{j1}^{(k)}=\sum _{d\le D}\xi _{d}^{+}\sum _{m\in \mathcal {M}_{k}}\sum _{1\le |h|\le H}c_{h}e\Big (-\frac{h}{d}(N+j-[m^{c}])^{\theta }\Big ) \end{aligned}$$

and

$$\begin{aligned} R_{j2}^{(k)}=\sum _{d\le D}\xi _{d}^{+}\sum _{m\in \mathcal {M}_{k}}\Delta _{H}\Big (-\frac{h}{d}(N+j-[m^{c}])^{\theta }\Big ). \end{aligned}$$

Let

$$\begin{aligned} W_{j}(v)=\sum _{m\in \mathcal {M}_{k}}e(v(N+j-[m^{c}])^{\theta }). \end{aligned}$$

Changing the order of summation together with Lemma 2.3, we obtain

$$\begin{aligned} R_{j1}^{(k)}\ll \sum _{d\le D}\sum _{1\le h\le H}\frac{1}{h}\left| W_{j}\Big (\frac{h}{d}\Big )\right| , \end{aligned}$$
(3.11)

and

$$\begin{aligned} R_{j2}^{(k)}\ll \sum _{d\le D}\frac{W(0)}{H}+\sum _{d\le D}\sum _{1\le h\le H}\frac{1}{H}\left| W_{j}\Big (\frac{h}{d}\Big )\right| . \end{aligned}$$
(3.12)

Hence, by (3.3), we obtain

$$\begin{aligned} R_{j2}^{(k)}\ll \frac{1}{\log N}\sum _{d\le D}\frac{N^{\theta }}{H}+\sum _{d\le D}\sum _{1\le h\le H}\frac{1}{H}\left| W_{j}\Big (\frac{h}{d}\Big )\right| . \end{aligned}$$

We choose

$$\begin{aligned} H=dN^{1-\theta }(\log N)^{3}. \end{aligned}$$

Combining (3.10), (3.11) and (3.12), we obtain

$$\begin{aligned} R_{j}^{(k)}\ll \frac{N^{2\theta -1}}{(\log N)^{3}}+\sum _{d\le D}\sum _{1\le h\le H}\frac{1}{h}\left| W_{j}\Big (\frac{h}{d}\Big )\right| ,\ \ j=0,1. \end{aligned}$$
(3.13)

Now, we consider the sum \(W_{j}(v)\). From (3.7) it follows that

$$\begin{aligned} W_{j}(v)\ll \sum _{m\in \mathcal {M}_{k}}e(v(N+j-[m^{c}])^{\theta })\sum _{s=0}^{2Z-1}g_{s}(m^{c})=\sum _{s=0}^{2Z-1}W_{j}^{(s)}(v), \end{aligned}$$

where

$$\begin{aligned} W_{j}^{(s)}(v)=\sum _{m\in \mathcal {M}_{k}}g_{s}(m^{c})e(v(N+j-[m^{c}])^{\theta }). \end{aligned}$$

By (3.2), (3.8) and (3.9). we have

$$\begin{aligned} W_{j}^{(0)}(v)\ll&\sum _{m\in \mathcal {M}_{k}}g_{0}(m^{c})\le \sum _{AN^{\theta }-1\le m<BN^{\theta }+1}g_{0}(m^{c})\nonumber \\ \ll&\frac{N^{\theta }}{Z}+\left| \sum _{AN^{\theta }-1\le m<BN^{\theta }+1}\sum _{1\le |n|\le Z(\log N)^{4}}\beta _{n}e(nm^{c})\right| +1\nonumber \\ \ll&\frac{N^{\theta }}{Z}+\frac{1}{Z}\sum _{1\le |n|\le Z(\log N)^{4}}|H_{n}|+1, \end{aligned}$$
(3.14)

where \(A=(1-(2\delta )^{c})^{\theta },\) \(B=(1-\delta ^{c})^{\theta }\) and

$$\begin{aligned} H_{n}=\sum _{AN^{\theta }-1\le m<BN^{\theta }+1}e(nm^{c}). \end{aligned}$$

Let \(\vartheta _{n}(x)=nx^{c}.\) It is easy to verify that \(\vartheta _{n}^{\prime \prime }(x)\asymp nN^{1-2\theta }\) uniformly for \(AN^{\theta }-1\le x<BN^{\theta }+1.\) Hence, by Lemma 2.4, we get

$$\begin{aligned} H_{n}\ll N^{\theta }(nN^{1-2\theta })^{1/2}+(nN^{1-2\theta })^{-1/2}\ll (nN)^{1/2}. \end{aligned}$$
(3.15)

We assume that

$$\begin{aligned} Z\ll N^{(2\theta -1)/3}(\log N)^{-4}. \end{aligned}$$
(3.16)

Then by (3.14), (3.15) and (3.16), we obtain

$$\begin{aligned} W_{j}^{(0)}(v)\ll \frac{N^{\theta }}{Z}+N^{1/2}Z^{1/2}\log ^{6} N\ll \frac{N^{\theta }}{Z}. \end{aligned}$$
(3.17)

Now, we consider the sums \(W_{j}^{(s)}(v)\) for \(1\le s\le 2Z-1.\) From (3.5) we know that \(g_{s}(m^{c})\) vanishes unless \(\{m^{c}\}\in [(s-1)/(2Z),(s+1)/(2Z)].\) Hence, the only summands in the sums \(W_{s}(v)\) are those for which

$$\begin{aligned} \{m^{c}\}=\frac{s}{2Z}+O\Big (\frac{1}{Z}\Big ). \end{aligned}$$

And in this case we have

$$\begin{aligned} v(N+j-[m^{c}])^{\theta }=v\Big (N+j-m^{c}+\frac{s}{2Z}\Big )^{\theta }+O\Big (\frac{vN^{\theta -1}}{Z}\Big ) \end{aligned}$$

and so

$$\begin{aligned} e(v(N+j-[m^{c}])^{\theta })=e\Big (v\Big (N+j-m^{c}+\frac{s}{2Z}\Big )^{\theta }\Big )+O\Big (\frac{vN^{\theta -1}}{Z}\Big ). \end{aligned}$$

Hence, we have

$$\begin{aligned} W_{j}^{(s)}(v)=V_{j}^{(s)}(v)+O\Big (\frac{vN^{\theta -1}}{Z}\sum _{m\in \mathcal {M}_{k}}g_{s}(m^{c})\Big ), \end{aligned}$$

where

$$\begin{aligned} V_{j}^{(s)}(v)=\sum _{m\in \mathcal {M}_{k}}g_{s}(m^{c})e\Big (v\Big (N+j-m^{c}+\frac{s}{2Z}\Big )^{\theta }\Big ). \end{aligned}$$
(3.18)

Thus, by (3.17), we get

$$\begin{aligned} W_{j}(v)=\sum _{s=1}^{2Z-1}W_{j}^{(s)}(v)+W_{j}^{(0)}(v)=\sum _{s=1}^{2Z-1}V_{j}^{(s)}(v)+O(\Xi )+O\Big (\frac{N^{\theta }}{Z}\Big ), \end{aligned}$$

where

$$\begin{aligned} \Xi =\frac{vN^{\theta -1}}{Z}\sum _{m\in \mathcal {M}_{k}}\sum _{s=1}^{2Z-1}g_{s}(m^{c}). \end{aligned}$$

By (3.3), (3.5) and (3.6), we have

$$\begin{aligned} \Xi \ll \frac{vN^{2\theta -1}}{Z\log N}. \end{aligned}$$

Now, we have

$$\begin{aligned} W_{j}(v)=\sum _{s=1}^{2Z-1}V_{j}^{(s)}(v)+O\Big (\frac{vN^{2\theta -1}}{Z\log N}+\frac{N^{\theta }}{Z}\Big ). \end{aligned}$$
(3.19)

Take

$$\begin{aligned} v=\frac{h}{d},\ \ \text{ where }\ \ 1\le d\le D,\ \ 1\le h\le H=dN^{1-\theta }(\log N)^{3}. \end{aligned}$$

Obviously, we have \(vN^{2\theta -1}\ll N^{\theta }(\log N)^{3}.\) So we can rewrite (3.19) as

$$\begin{aligned} W_{j}(v)=\sum _{s=1}^{2Z-1}V_{j}^{(s)}(v)+O\Big (\frac{N^{\theta }}{Z}(\log N)^{2}\Big ). \end{aligned}$$
(3.20)

Combining (3.13) and (3.20), we obtain

$$\begin{aligned} R_{j}^{(k)}\ll \frac{N^{2\theta -1}}{(\log N)^{3}}+\sum _{d\le D}\sum _{1\le h\le H}\frac{1}{h}\sum _{s=1}^{2Z-1}\left| V_{j}^{(s)}\Big (\frac{h}{d}\Big )\right| +\sum _{d\le D}\sum _{1\le h\le H}\frac{1}{h}\frac{N^{\theta }}{Z}(\log N)^{2}. \end{aligned}$$

We choose Z such that

$$\begin{aligned} Z\asymp dN^{1-\theta }(\log N)^{7}. \end{aligned}$$

Hence, we have

$$\begin{aligned} R_{j}^{(k)}\ll \frac{N^{2\theta -1}}{(\log N)^{3}}+\sum _{d\le D}\sum _{1\le h\le H}\frac{1}{h}\sum _{s=1}^{2Z-1}\left| V_{j}^{(s)}\Big (\frac{h}{d}\Big )\right| . \end{aligned}$$
(3.21)

Now, we consider the sums \(V_{j}^{(s)}(h/d).\) By (3.8) and (3.18), we have

$$\begin{aligned} V_{j}^{(s)}\Big (\frac{h}{d}\Big )&=\sum _{m\in \mathcal {M}_{k}}\Bigg (\sum _{|n|\le Z(\log N)^{4}}\beta ^{(s)}_{n}e(nm^{c})\Bigg )e \Big (v\Big (N+j-m^{c}+\frac{s}{2Z}\Big )^{\theta }\Big )+O(N^{-10})\nonumber \\&=\sum _{|n|\le Z(\log N)^{4}}\beta ^{(s)}_{n}U\Big (N+j+\frac{s}{2Z},n,\frac{h}{d}\Big )+O(N^{-10})\nonumber \\&\ll \frac{1}{Z}\sum _{|n|\le R}\sup _{T\in [N,N+2]}\left| U\left( T,n,\frac{h}{d}\right) \right| , \end{aligned}$$
(3.22)

where

$$\begin{aligned} U=U(T,n,v)=\sum _{m\in \mathcal {M}_{k}}e(nm^{c}+v(T-m^{c})^{\theta }) \ \ \text{ and }\ \ R=dN^{1-\theta }(\log N)^{12}. \end{aligned}$$

Inserting (3.22) into (3.21), we get

$$\begin{aligned} R_{0}^{(k)}+R_{1}^{(k)}\ll \frac{N^{2\theta -1}}{(\log N)^{3}}+\sum _{d\le D}\sum _{1\le h\le H}\frac{1}{h}\sum _{|n|\le R}\sup _{T\in [N,N+2]}\left| U\Big (T,n,\frac{h}{d}\Big )\right| . \end{aligned}$$
(3.23)

Recall the definition of \(\mathcal {M}_{k}\) in (3.2). Let \(k\ge 2\) and \(n=p_{1}\cdots p_{k}\in \mathcal {M}_{k}.\) By some routine arguments, we can rewrite n as \(n=rs\) with

$$\begin{aligned} N^{\frac{\theta }{200}}<r\le N^{\frac{1}{2}}<s<N^{\frac{199\theta }{200}}. \end{aligned}$$

In fact, it is easy to see that \(U\Big (T,n,\frac{h}{d}\Big )\) is a summation similar to (121) in [11]. Through the same argument as Sect. 3.6 of [11], there is almost no need for adjustment, and we can get that if

$$\begin{aligned} \frac{28}{29}<\theta<1,\ \ \delta <\frac{29\theta -28}{26}, \end{aligned}$$

then

$$\begin{aligned} \sum _{d\le D}\sum _{1\le h\le H}\frac{1}{h}\sum _{|n|\le R}\sup _{T\in [N,N+2]}\left| U\Big (T,n,\frac{h}{d}\Big )\right| \ll \frac{N^{2\theta -1}}{(\log N)^{3}}. \end{aligned}$$

So, we have

$$\begin{aligned} R_{0}^{(k)}+R_{1}^{(k)}\ll \frac{N^{2\theta -1}}{(\log N)^{3}}. \end{aligned}$$

This completes the proof. \(\square \)

4 The Proof of Theorem 1.1

To prove the theorem, we consider the lower bound of the sum

$$\begin{aligned} \Gamma ={\mathop {\sum }\limits _{\begin{array}{c} P<p\le 2P,m\in \mathbb {N}\\ {[}p^{c}]+[m^{c}]=N\\ m=P_{10} \end{array}}}(\log p). \end{aligned}$$

By the trivial inequality

$$\begin{aligned} \Gamma \ge&\mathop {\sum }\limits _{\begin{array}{c} P<p\le 2P,\ m\in \mathbb {N}\\ {[}p^{c}]+[m^{c}]=N\\ (m,P(z))=1 \end{array}}(\log p)-\mathop {\sum }\limits _{\begin{array}{c} P<p\le 2P,\ m\in \mathbb {N}\\ {[}p^{c}]+[(p_{1}\ldots p_{10}m)^{c}]=N\\ z\le p_{1}\le \ldots \le p_{10},(m,P(p_{10})=1) \end{array}}(\log p)\nonumber \\ \ge&\mathop {\sum }\limits _{\begin{array}{c} P<p\le 2P,\ m\in \mathbb {N}\\ {[}p^{c}]+[m^{c}]=N\\ (m,P(z))=1 \end{array}}(\log p)-\mathop {\sum }\limits _{\begin{array}{c} \ell ,m\in \mathbb {N},\ P<\ell \le 2P\\ {[}\ell ^{c}]+[(p_{1}\ldots p_{10}m)^{c}]=N\\ z\le p_{1}\le \ldots \le p_{10},(m,P(p_{10})=1)\\ (\ell ,P(z))=1 \end{array}}(\log \ell )\nonumber \\ =&\Gamma _{1}-\Gamma _{2}. \end{aligned}$$
(4.1)

From (6), (17), (18) and (19) in [11], we know that

$$\begin{aligned} \Gamma _{1}\ge \Sigma +\Sigma _{0}-\Sigma _{1}, \end{aligned}$$

where \(\Sigma _{j}, j=0,1\) are defined by (2.4) and

$$\begin{aligned} \Sigma \ge A(N)V(z)(f(s)+o(1)) \end{aligned}$$

with

$$\begin{aligned} A(N)=\theta \sum _{P<p\le 2P}(\log p)((N-[p^{c}])^{\theta -1}+O(N^{\theta -2})) \end{aligned}$$

and

$$\begin{aligned} s=\frac{\log D}{\log z}. \end{aligned}$$

By Lemma 2.2 and Proposition 3.1, we can take

$$\begin{aligned} \eta =\frac{29\theta -28}{26}-\varepsilon . \end{aligned}$$

So we have

$$\begin{aligned} s=\frac{200(29\theta -28)}{26}+o(1). \end{aligned}$$
(4.2)

From the definition of P and the prime number theorem, we obtain

$$\begin{aligned} A(N)\ge (\delta \theta (1-(2\delta )^{c})^{\theta -1}+o(1))N^{2\theta -1}. \end{aligned}$$

Hence, by Lemma 2.2 and the fact

$$\begin{aligned} V(z)=\prod _{p<z}\left( 1-\frac{1}{p}\right) \asymp \frac{1}{\log z}\asymp \frac{1}{\log N}, \end{aligned}$$
(4.3)

we get

$$\begin{aligned} \Gamma _{1}\ge (\theta (\delta (1-(2\delta )^{c})^{\theta -1})+o(1))N^{2\theta -1}V(z)(f(s)+o(1)). \end{aligned}$$
(4.4)

Obviously, we have

$$\begin{aligned} \Gamma _{2}\le \sum _{k=11}^{199}{\mathop {\mathop {\mathop {\sum }\limits _{\ell \in \mathbb {N}}}\limits _{[\ell ^{c}]+[m^{c}]=N}}\limits _{ m\in \mathcal {M}_{k},\ (\ell ,P(z))=1}}(\log \ell )=(1+o(1))\sum _{k=11}^{199}(\log P)\Sigma _{2}^{k}, \end{aligned}$$
(4.5)

where

$$\begin{aligned} \Sigma _{2}^{k}={\mathop {\mathop {\mathop {\sum }\limits _{\ell \in \mathbb {N}}}\limits _{[\ell ^{c}]+[m^{c}]=N}}\limits _{m\in \mathcal {M}_{k},\ (\ell ,P(z))=1}}1. \end{aligned}$$

From (2.1), we find

$$\begin{aligned} \Sigma _{2}^{k}={\mathop {\mathop {\mathop {\sum }\limits _{\ell \in \mathbb {N}}}\limits _{[\ell ^{c}]+[m^{c}]=N}} \limits _{m\in \mathcal {M}_{k}}}\sum _{d\mid (\ell ,P(z))}\mu (d)\le {\mathop {\mathop {\mathop {\sum }\limits _{\ell \in \mathbb {N}}}\limits _{[\ell ^{c}]+[m^{c}]=N}} \limits _{m\in \mathcal {M}_{k}}}\sum _{d\mid (\ell ,P(z))}\xi ^{+}(d). \end{aligned}$$

By exchanging the order of summation, we obtain

$$\begin{aligned} \Sigma _{2}^{k}\le \sum _{d\mid P(z)}\xi ^{+}(d)G_{d,k},\ \ \text{ where }\ \ G_{d,k}={\mathop {\mathop {\mathop {\sum }\limits _{\ell \in \mathbb {N}}} \limits _{[\ell ^{c}]+[m^{c}]=N}} \limits _{m\in \mathcal {M}_{k},\ \ell \equiv 0\ (\mathrm{{mod}}\ d)}}1. \end{aligned}$$
(4.6)

By the identity

$$\begin{aligned} \sum _{a\le m<b}1=[-a]-[-b]=b-a-\rho (-b)+\rho (-a), \end{aligned}$$

we have

$$\begin{aligned} G_{d,k}&=\sum _{m\in \mathcal {M}_{k}}{\mathop {\mathop {\mathop {\sum }\limits _{\ell \in \mathbb {N}}} \limits _{[\ell ^{c}]+[m^{c}]=N}} \limits _{\ell \equiv 0\ (\mathrm{{mod}}\ d)}}1=\sum _{m\in \mathcal {M}_{k}}\sum _{(1/d)(N-[m^{c}])^{\theta }\le \ell <(1/d)(N+1-[m^{c}])^{\theta }}1\nonumber \\&=\sum _{m\in \mathcal {M}_{k}}\frac{(N+1-[m^{c}])^{\theta }-(N-[m^{c}])^{\theta }}{d}\nonumber \\&\quad +\sum _{m\in \mathcal {M}_{k}}\rho \Big (-\frac{1}{d}(N-[m^{c}])^{\theta }\Big )-\sum _{m\in \mathcal {M}_{k}}\rho \Big (-\frac{1}{d}(N+1-[m^{c}])^{\theta }\Big ). \end{aligned}$$
(4.7)

Combining (4.6), (4.7) and the identity

$$\begin{aligned} (N+1-[m^{c}])^{\theta }=(N-[m^{c}])^{\theta }+\theta (N-[m^{c}])^{\theta -1}+O(N^{\theta -2}), \end{aligned}$$

we obtain

$$\begin{aligned} \Sigma _{2}^{k}\le R^{(k)}+R^{(k)}_{0}-R^{(k)}_{1}, \end{aligned}$$
(4.8)

where \(R^{(k)}_{j}, j=0,1\) are defined by (3.1) and

$$\begin{aligned} R^{(k)}=\theta \sum _{d\mid P(z)}\frac{\xi ^{+}(d)}{d}\sum _{m\in \mathcal {M}_{k}}((N-[m^{c}])^{\theta -1}+O(N^{\theta -2})). \end{aligned}$$

By (2.2) and (3.2), we have

$$\begin{aligned} R^{(k)}\le \theta (2\delta )^{1-c}N^{\theta -1}\Big (\sum _{m\in \mathcal {M}_{k}}1\Big )V(z)(F(s)+o(1)), \end{aligned}$$
(4.9)

where

$$\begin{aligned} s=\frac{200(29\theta -28)}{26}+o(1). \end{aligned}$$

By (3.4), (4.3), (4.5), (4.8), (4.9) and Proposition 3.1, we have

$$\begin{aligned} \Gamma _{2}\le (0.840654+o(1))\theta (2\delta )^{1-c}((1-\delta ^{c})^{\theta }-(1-(2\delta )^{c})^{\theta })F(s)V(z)N^{2\theta -1}. \end{aligned}$$
(4.10)

From (4.1), (4.4) and (4.10), as long as

$$\begin{aligned} L= & {} (1-(2\delta )^{c})^{\theta -1}f(s)-0.840654\delta ^{-1}(2\delta )^{1-c}((1-\delta ^{c})^{\theta }\\{} & {} -(1-(2\delta )^{c})^{\theta })F(s)>0, \end{aligned}$$

we can deduce that

$$\begin{aligned} \Gamma \gg \frac{N^{2\theta -1}}{\log N}, \end{aligned}$$

which leads to the theorem. Recall that \(\delta =10^{-9}.\) By (4.2) and Lemma 2.1, one can use the software Mathematica to run the following code, which shows that \(L[1.0198]>0.0017884.\)

F[x_] := Piecewise [{{(2E\(^\wedge \)EulerGamma)/x, 0 < x \(<=\) 3}, {((2E\(^\wedge \)EulerGamma)/x) * (1 + NIntegrate [Log [t \(-1\)]/t, {t, 2, x \(-1\)}]), 3 \(<=\) x < 5}, {((2E\(^\wedge \)EulerGamma)/x) * (1 + NIntegrate [ Log[t \(-1\)]/t, {t, 2, x \(-1\)}] + NIntegrate [(Log [t \(-1\)]/(t * u)) * Log [(u \(-1\))/(t \(+1\))], {t, 2, x \(-3\)}, {u, t+2, x \(-1\)}]), 5 \(<=\) x<7}}]

f[x_] := Piecewise [{{((2E\(^\wedge \)EulerGamma)/x) * Log [x \(-1\)], 2 \(<=\) x \(<=\) 4}, {((2E\(^\wedge \)EulerGamma) /x) * (Log [x \(-1\)] + NIntegrate [Log [u \(-1\)]/(t * u), {t, 3, x \(-1\)}, {u, 2, t \(-1\)}]), 4 \(<=\) x < 6}, {((2E\(^\wedge \)Eu lerGamma)/x) * (Log [x \(-1\)] + NIntegrate [Log [u \(-1\)]/(t * u), {t, 3, x \(-1\)}, {u, 2, t \(-1\)}] + NIntegrate [ (Log [t \(-1\)]/(t * u)) * Log [(u \(-1\))/(t + 1)] * Log[x/(u + 2)], {t, 2, x \(-4\)}, {u, t + 2, x \(-2\)}]), 6 \(<=\) x \(<=\) 8}} ]

L[x_] := ((1 − (2 * 10\(^\wedge \)(− 9))\(^\wedge \)x)\(^\wedge \) (1/x \(-1\))) * f [(200 * (29–28x))/(26x)] − (10\(^\wedge \)9) * (2 * 10\(^\wedge \)(− 9))\(^\wedge \) (1 −x) ((1–10\(^\wedge \)(− 9x))\(^\wedge \) (1/x) − (1 − (2 * 10\(^\wedge \)(− 9))\(^\wedge \)x)\(^\wedge \) (1/x)) F [(200 * (29–28x))/(26x)] * 0.840654

From [5, Chapter 11], we know that f(s) is an increasing function and F(s) is a decreasing function. By a trivial argument, we can conclude that L is a decreasing function about c, which deduces that if \(1<c<1.0198,\) then \(L>0.0017884\). This completes the proof of the theorem.