Distribution Modulo One of αp^γ + β for Special Classes of Primes

Abstract

Let $\alpha ,\beta \in \mathbb{R}$ with $\alpha \ne 0$, and let $\gamma \in (0,5/6)$. Define the set $M_1$ to consist of primes p such that $p+2$ is almost prime, and let $M_2$ be the set of primes of the form $p=a^2+b^2+1$. We study the distribution of $\alpha p^{\gamma }+\beta$ modulo one, as p ranges over the sets $M_1$ and $M_2$, respectively.

1. Introduction and Statements of the Result

The problem of the distribution of fractional parts of the sequence $\alpha n^k$ was first considered by Hardy and Littlewood [1]. Later, Vinogradov (see Chapter 11 of [2,3]) studied the distribution of the fractional parts of $\alpha p^k$ with p—prime for the cases $k=1$ and $k\ge 2$. The case when the exponent of the prime is not an integer is also important. One of Landau’s four problems, which were presented at the 1912 International Congress of Mathematicians, is the problem of having infinitely many primes of the form $n^2+1$. This problem is still unsolved, and it is equivalent to the existence of infinitely many primes p such that

$$\left\{\sqrt{p}\right\}<p^{-\theta }$$

(1)

with $\theta =1/2$ (as usual $\left\{x\right\}=x-\left[x\right]$, where $\left[x\right]$ is the largest integer not greater than x). The best exponent $\theta =0.262+\epsilon$ for the right-hand side of (1) is due to Harman and Lewis [4], but their result is applicable only for $\left\{p^{\lambda }\right\}$ with $\lambda =1/2$. For $\lambda \in (0,1)$ Balog and Harman obtain non-trivial results—see [5,6,7,8,9]. Subsequently, Baier [10] obtained asymptotic formulae for the number of primes in an interval that satisfies $\{p^{\lambda }-\theta \}<\delta$ with $\delta$ depending on a fractional power of p.

Another still unproven conjecture states that infinitely many primes p exist such that $p+2$ is also a prime. There are several established approximations for it. In 1973, Chen [11] proved that there are infinitely many primes p for which $p+2={\mathrm{P}}_2$. As usual, we denote by ${\mathrm{P}}_r$ an integer with no more than r prime factors, counted according to multiplicity.

In 2013, Cai [12] mixed these two problems and proved the existence of infinitely many primes p such that

$$\left\{\sqrt{p}\right\}<p^{-\theta }\mathrm{and}p+2=P_r$$

(2)

with $\theta =1/15.5$ and $r=4$. Later Li [13] (2023) and [14] (2025) improve Cai’s result. Let $M(\theta ,r)$ denote that the conditions (2) are fulfilled for infinitely many primes p. Using this notation, we can state the results of Cai [12] and Li [13,14] in the following form:

$$\begin{array}{cc}\hfill & M(\frac{1}{15.5},4),\hfill \\ \hfill & M(\theta ,\lfloor {\displaystyle \frac{8}{1-4\theta }}\rfloor )\mathrm{with}\theta \in (0,1/4),\hfill \\ \hfill & M(\frac{1}{15.1},4),\hfill \\ \hfill & M(\frac{1}{9.83},5),\hfill \\ \hfill & M(\frac{1}{9.38},6),\hfill \\ \hfill & M(\frac{1}{9.34},7).\hfill \end{array}$$

Meanwhile, in 2017, Dunn [15] proved that if $0<\gamma <1$ and $\theta <\gamma /10$ then there are infinitely many primes p such that

$$\left|\right|\alpha p^{\gamma }+\beta \left|\right|<p^{-\theta }\mathrm{and}p+2=P_3.$$

Here, following standard notation, we denote by $\left|\right|x\left|\right|=\underset{n\in \mathbb{Z}}{min}|x-n|$ the distance from x to the nearest integer. Our first statement improves Dunn’s result.

Theorem 1. 

Let $\alpha \in \mathbb{R}\setminus \left\{0\right\}$, $\beta \in \mathbb{R}$, $0<\gamma <5/6$ and $\theta <\gamma /10$. Then there are infinitely many primes p such that

$$\left|\right|\alpha p^{\gamma }+\beta \left|\right|<p^{-\theta }\mathrm{and}p+2=P_2.$$

(3)

If we assume slightly stronger restrictions on $\gamma$ and $\theta$, then we get

Theorem 2. 

Let α and β be defined as above and

$$4\theta <\gamma <\frac{5}{6}-\theta .$$

(4)

Then there are infinitely many primes p satisfying (3).

From the above theorem, we directly obtain the following corollary.

Corollary 1. 

There are infinitely many primes p such that

$$\left|\right|\sqrt{p}\left|\right|<p^{-1/8}$$

and $p+2=P_2$.

This result extends the work of Cai [12] and Li [14] in relation to the distance-to-the-nearest-integer function.

Our second result is connected to Linnik’s primes. In 1960, Linnik [16] showed that there exist infinitely many prime numbers of the form $p=a^2+b^2+1$. He proved the asymptotic formula

$$\sum _{p\le X}r(p-1)=\pi \prod _{p>2}\left(1+{\displaystyle \frac{\chi \left(p\right)}{p(p-1)}}\right){\displaystyle \frac{X}{logX}}+\mathcal{O}\left({\displaystyle \frac{X{(loglogX)}^5}{{(logX)}^{1+{\theta }_0}}}\right),$$

where $r\left(n\right)$ is the number of representations of n as a sum of two squares, $\chi \left(n\right)$ is the non-principal character modulo 4, and ${\theta }_0={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{4}}elog2$.

In 2017 Teräväinen [17] proved that there are infinitely many primes p such that $p=a^2+b^2+1$ and $\left|\right|\alpha p+\beta \left|\right|<p^{-1/80+\epsilon }$, where $\alpha$ is irrational.

We couple the problem of the distribution of $\alpha p^{\gamma }+\beta$ modulo one and the existence of Linnik’s primes, and we prove

Theorem 3. 

Let $\alpha \in \mathbb{R}\setminus \left\{0\right\}$, $\beta \in \mathbb{R}$, $0<\gamma <5/6$ and $\theta <\gamma /10$. Then there are infinitely many primes p of the form $p=a^2+b^2+1$ and such that

$$\left|\right|\alpha p^{\gamma }+\beta \left|\right|<p^{-\theta }.$$

2. Notation

Let x be a sufficiently large real number,

$$\begin{array}{cc}\hfill \Delta & =\Delta \left(x\right)=x^{-\theta },\hfill \\ \hfill K& ={\Delta }^{-1}{log}^{2}x,\hfill \\ \hfill z& =x^{1/\delta },\hfill \end{array}$$

(5)

where $\theta$ and $\delta$ are parameters, which will be chosen later, and they will be different in the proofs of Theorems 1 and 2.

By $q,p,p_1,p_2$, we denote prime numbers and by $\phi \left(n\right),\mu \left(n\right),\mathsf{\Lambda }\left(n\right)$, and $\tau \left(n\right)$ we denote the Euler totient function, the Möbius function, Mangoldt’s function, and the number of solutions to the equation $m_1{m}_2=n$ in natural numbers $m_1,m_2$. Let $(m_1,\dots ,m_k)$ and $[m_1,\dots ,m_k]$ be the largest common divisor and the least common multiple of $m_1,\dots ,m_k$, respectively. Instead of $m\equiv n(modk)$, for simplicity, we write $m\equiv n\left(k\right)$. We denote $e\left(y\right)=e^{2\pi iy}$ and $n\sim x$ as means that n runs through a subinterval of $(x,2x]$ with endpoints that are not necessarily the same in the different equations. The letter $\epsilon$ denotes an arbitrary small positive number, not the same in all appearances. For example, this convention allows us to write $x^{\epsilon }logx\ll x^{\epsilon }$.

3. Auxiliary Results

This section gives some auxiliary results for the reader’s convenience. We will use Bombieri–Vinogradov’s theorem and its strengthened form, proven by Bombieri, Friedlander, and Iwaniec:

Theorem 4 

([18], ch. 24, Bombieri–Vinogradov). For every $0<\theta <\frac{1}{2}$, $D=x^{\theta }$, and $A>0$, we have

$$\sum _{d\le D}\underset{(a,d)=1}{max}|\pi (x,d,a)-{\displaystyle \frac{\pi \left(x\right)}{\phi \left(d\right)}}|{\ll }_A{\displaystyle \frac{x}{{(logx)}^{A-5}}}$$

Theorem 5 

([19], Theorem 10). Let $\epsilon >0$, $D=x^{4/7-\epsilon }$ and $\lambda (.)$ denote a well-factorable function of level D. Then, for any given $A>0$ and $a\ne 0$, we have

$$\sum _{\begin{array}{c}q\le D\\ (d,a)=1\end{array}}\lambda \left(d\right)\left(\pi (x,d,a)-{\displaystyle \frac{\pi \left(x\right)}{\phi \left(d\right)}}\right){\ll }_A{\displaystyle \frac{x}{{(logx)}^A}}.$$

The constant implied in ≪ depends at most on $\epsilon ,a$ and A.

The following Lemma is a generalization of Bombieri–Vinogradov’s theorem:

Theorem 6 

([20]). Let $g(x,k)$ be an arithmetic function satisfying $g(x,k)\ll {\tau }_r\left(k\right)$, $1/2\le E\left(x\right)\ll x^{1-\theta }$, $0<\theta \le 1$ and

$$H(y;k,d,l)=\sum _{\begin{array}{c}kp\le y\\ kp\equiv l\left(d\right)\end{array}}1-{\displaystyle \frac{1}{\varphi \left(d\right)}}\sum _{kp\le y}1$$

Then for any given constant $A>10$, there exists a constant $B=B\left(A\right)>0$ such that for $D\le {\displaystyle \frac{x^{1/2}}{{(logx)}^B}}$ the inequality

$$\sum _{\begin{array}{c}d\le D\end{array}}\underset{(d,l)=1}{max}\underset{y\le x}{max}\left|\sum _{\begin{array}{c}k\le E\left(x\right)\\ (k,d)=1\end{array}}g(x,k)H(y;k,d,l)\right|\ll {\displaystyle \frac{x}{{log}^{A}x}}$$

is fulfilled.

Our primary tool will be a half-dimensional linear sieve. We will use the following notation.

Let $\mathcal{P}$ be a set of prime numbers,

$$P\left(z\right)=\prod _{\begin{array}{c}p\le z\\ p\in \mathcal{P}\end{array}}p.$$

For the set $\mathcal{A}=\left\{a_n\right|n\le x\}$ of nonnegative numbers, we will impose the following conditions:

C1.

if ${\mathcal{A}}_d=\sum _{\begin{array}{c}n\le x\\ d|n\end{array}}{a}_n$ then there exists multiplicative function $\omega \left(d\right)$ such that

$$|{\mathcal{A}}_d|={\displaystyle \frac{\omega \left(d\right)}{d}}X+r(\mathcal{A},d)$$

(6)

where X is independent of d and $r(\mathcal{A},d)$ is a real number considered as an error term.

C2.

The multiplicative function $\omega \left(d\right)$ satisfies $0\le \omega \left(p\right)<p$ and

$$\prod _{\begin{array}{c}{z}_1\le p<z_2\\ p\in \mathcal{P}\end{array}}{\left(1-{\displaystyle \frac{\omega \left(p\right)}{p}}\right)}^{-1}\le {\left({\displaystyle \frac{log{z}_2}{log{z}_1}}\right)}^{\kappa }\left(1+{\displaystyle \frac{L}{log{z}_1}}\right)$$

(7)

for all $z_2>z_1\ge 2$ with some constant $\kappa \ge 0$ and absolute constant $L\ge 1$.

Definition 1. 

The arithmetical functions ${\lambda }^{\pm }$ are called upper and lower bound sieve weights of level $D\ge 2$ for the set of primes $\mathcal{P}$ if

D1. 

for any positive integer d we have

$$|{\lambda }^{\pm }\left(d\right)|\le 1\mathrm{if}d|P\left(z\right)\mathrm{and}{\lambda }^{\pm }\left(d\right)=0\mathrm{if}d>D\mathrm{or}\mu \left(d\right)=0;$$

D2. 

for $n\in \mathbb{N}$ the inequalities

$$\sum _{d|n}{\lambda }^{-}\left(d\right)\le \sum _{d|n}\mu \left(d\right)\le \sum _{d|n}{\lambda }^{+}\left(d\right).$$

are fulfilled.

We will use the following notation.

$$S(\mathcal{A},\mathcal{P},z)=\sum _{\begin{array}{c}n\in \mathcal{A}\\ (n,P(z\left)\right)=1\end{array}}{a}_n\mathrm{and}V\left(z\right)=\prod _{\begin{array}{c}p\le z\\ p\in \mathcal{P}\end{array}}\left(1-{\displaystyle \frac{\omega \left(p\right)}{p}}\right).$$

(8)

We will use a linear sieve.

Lemma 1 

([21]). Suppose that for the set $\mathcal{A}$ the conditions (C1.) and (C2.) are fulfilled and the inequality (7) is fulfilled with $\kappa =1$. Then, there exists arithmetical functions ${\lambda }^{\pm }\left(d\right)$ (called Rosser’s weights of level D) with the properties (D1.) (D2.) and for $s={\displaystyle \frac{logD}{logz}}$ we have

$$\begin{array}{cc}\hfill S(\mathcal{A},z)\le & XV(z)(F(s)+O({(logD)}^{-{\displaystyle \frac{1}{3}}}))\hfill \\ \hfill & +\sum _{d\le D}{\lambda }^{+}\left(d\right)r(\mathcal{A},d)\mathrm{if}2\le z\le D,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill S(\mathcal{A},z)\ge & V(z)X(f(s)+O({(logD)}^{-{\displaystyle \frac{1}{3}}}))\hfill \\ \hfill & +\sum _{d\le D}{\lambda }^{-}\left(d\right)r(\mathcal{A},d)2\le z\le D^{1/2},,\hfill \end{array}$$

(10)

where $F\left(s\right)$ and $f\left(s\right)$ satisfy equations

$$\begin{array}{cc}\hfill & F\left(s\right)=2e^{\gamma }{s}^{-1},\mathrm{if}0<s\le 3,\hfill \\ \hfill & f\left(s\right)=2e^{\gamma }{s}^{-1}log(s-1),\mathrm{if}2\le s\le 3,\hfill \\ \hfill & {\left(sF\left(s\right)\right)}^{\prime }=f(s-1),\mathrm{if}s>3,\hfill \\ \hfill & {\left(sf\left(s\right)\right)}^{\prime }=F(s-1),\mathrm{if}s>2\hfill \end{array}$$

(11)

and γ is Euler’s constant. ($\gamma \approx 0.577$, also known as Euler–Mascheroni constant, is defined as the limiting difference between the harmonic series and the natural logarithm).

Also, we will use a linear sieve in the following form due to Iwaniec:

Lemma 2 

([22]). Suppose that for the set $\mathcal{A}$ the conditions (C1.) and (C2.) are fulfilled, the inequality (7) is fulfilled with $\kappa =1$, $2\le z\le D^{1/2}$ and $s={\displaystyle \frac{logD}{logz}}$. Then

$$S(\mathcal{A},z)\le xV(z)(F(s)+o(1))+\sum _{l\le L}\sum _{d|P(z)}{\lambda }_l^{+}(d)r(\mathcal{A},d)$$

and

$$S(\mathcal{A},z)\ge xV(z)(f(s)-o(1))-\sum _{l\le \mathcal{L}}\sum _{d|P(z)}{\lambda }_l^{-}(d)r(\mathcal{A},d)$$

where $\mathcal{L}\ll 1$, ${\lambda }_l^{\pm }$ are well factorable functions of level D and the functions $F,f$ are defined by (11).

The next Lemma gives the explicit form of the functions of the linear sieve for $s\le 6$:

Lemma 3 

([23]). For functions $F\left(s\right)$ and $f\left(s\right)$ of the linear sieve the following inequalities

$$\begin{array}{cc}\hfill F\left(s\right)& ={\displaystyle \frac{2e^{\gamma }}{s}}\mathit{if}0<s\le 3;\hfill \\ \hfill F\left(s\right)& ={\displaystyle \frac{2e^{\gamma }}{s}}\left(1+{\int }_2^{s-1}{\displaystyle \frac{log(t-1)}{t}}dt\right)\mathit{if}3\le s\le 5;\hfill \\ \hfill f\left(s\right)& ={\displaystyle \frac{2e^{\gamma }log(s-1)}{s}},2\le s\le 4;\hfill \\ \hfill f\left(s\right)& ={\displaystyle \frac{2e^{\gamma }}{s}}\left(log(s-1)+{\int }_3^{s-1}{\displaystyle \frac{dt}{t}}{\int }_2^{t-1}{\displaystyle \frac{log(u-1)}{u}}du\right)\hfill \\ \hfill & \mathit{if}4\le s\le 6\hfill \end{array}$$

are fulfilled.

The next Lemma is a fundamental Lemma for the semi-linear sieve:

Lemma 4. 

Suppose that for the set $\mathcal{A}$ the conditions (C1.) and (C2.) are fulfilled, the inequality (7) holds with $\kappa =1/2$. Then, there exists an arithmetical functions ${\lambda }_{\mathrm{sem}}^{\pm }$ with the properties (D1.) (D2.) such that

$$\begin{array}{cc}\hfill S(\mathcal{A},\mathcal{P},z)& \ge XV\left(z\right)\left(f_{sem}\left(s\right)+O\left({(logD)}^{-1/6}\right)\right)-\sum _{\begin{array}{c}d\le D\end{array}}{\lambda }_{sem}^{-}\left(d\right)r(\mathcal{A},d),\hfill \\ \hfill S(\mathcal{A},\mathcal{P},z)& \le XV\left(z\right)\left(F_{sem}\left(s\right)+O\left({(logD)}^{-1/6}\right)\right)+\sum _{\begin{array}{c}d\le D\end{array}}{\lambda }_{sem}^{+}\left(d\right)r(\mathcal{A},d),\hfill \end{array}$$

where $s={\displaystyle \frac{logD}{logz}}$ and $f_{sem},F_{sem}$ are continuous functions which satisfy

$$\begin{array}{cc}\hfill & \sqrt{s}{F}_{sem}\left(s\right)=2\sqrt{e^{\gamma }/\pi }\mathrm{if}0<s\le 2,\hfill \\ \hfill & f_{sem}\left(s\right)=0\mathrm{if}0<s\le 1,\hfill \\ \hfill & 2\sqrt{s}{\left(\sqrt{s}{F}_{sem}\left(s\right)\right)}^{\prime }=f_{sem}(s-1),\hfill \\ \hfill & 2\sqrt{s}{\left(\sqrt{s}{f}_{sem}\left(s\right)\right)}^{\prime }=F_{sem}(s-1)\hfill \end{array}$$

where γ is the Euler constant.

Proof. 

The proof follows from [24] Theorem 11.12–Theorem 11.13 with $\beta =1$ and [24] Chapter 14 (pp. 275–276). □

Let

$$\begin{array}{cc}\hfill & z=x^{1/\delta },2\le \delta \le 6,\hfill \\ \hfill & P_3=\{p\mid p\equiv 3(mod4)\},\hfill \\ \hfill & P_3\left(z\right)=\prod _{\begin{array}{c}p\le z\\ p\in P_3\end{array}}p,\hfill \\ \hfill & \mathcal{A}=\{p-1\mid x<p\le 2x,p\equiv 3(mod8\left)\right\},\hfill \\ \hfill & S(\mathcal{A},P_3,z)=\sum _{{\displaystyle \genfrac{}{}{0pt}{}{p\in \mathcal{A}}{(p-1,P_3\left(z\right))=1}}}1.\hfill \end{array}$$

(12)

The following Lemma provides a lower bound for $S(\mathcal{A},P_3,z)$.

Lemma 5 

([25], Proposition 6). Under condition (12) the inequality

$$S(\mathcal{A},P_3,z)\ge \left(W\left(\delta \right)+o\left(1\right)\right){\displaystyle \frac{x}{{(logx)}^{3/2}}},$$

where

$$\begin{array}{cc}\hfill W& \left(\delta \right)={\displaystyle \frac{A{C}_3}{\sqrt{2}}}\underset{1}{\overset{\delta /2}{\int }}{\displaystyle \frac{dt}{\sqrt{t(t-1)}}};\hfill \end{array}$$

$$A={\displaystyle \frac{1}{2\sqrt{2}}}\prod _{\begin{array}{c}p\equiv 3\left(\mathrm{mod}4\right)\end{array}}{\left(1-{\displaystyle \frac{1}{p^2}}\right)}^{1/2};$$

(13)

$$C_3=\prod _{\begin{array}{c}p\equiv 3\left(\mathrm{mod}4\right)\end{array}}\left(1-{\displaystyle \frac{1}{{(p-1)}^2}}\right)$$

(14)

is fulfilled.

We will use the following

Lemma 6 

([25], Lemma 5). Let $z=x^{1/\delta }$, $2\le \delta <4$,

$$\mathcal{L}=\left\{l=n{p}_1|\begin{array}{c}n\le x^{1-2/\delta },p\mid n\Rightarrow p\equiv 1(mod4),\hfill \\ x^{1/\delta }\le p_1<{(x/n)}^{1/2},p_1\in {\mathcal{P}}_3\hfill \end{array}\right\}$$

and

$$h\left(n\right)=\prod _{p|n}{\left(1-{\displaystyle \frac{1}{p-1}}\right)}^{-1},C_1=\prod _{\begin{array}{c}p\equiv 1\left(\mathrm{mod}4\right)\end{array}}\left(1-{\displaystyle \frac{1}{{(p-1)}^2}}\right).$$

(15)

Then

$$\sum _{\ell \in \mathcal{L}}{\displaystyle \frac{h(\ell )}{\ell log(x/\ell )}}={\displaystyle \frac{1+o\left(1\right)}{{(logx)}^{1/2}}}·{\displaystyle \frac{A}{2C_1}}{\int }_2^{\delta }{\displaystyle \frac{log(t-1)}{t{(1-t/\alpha )}^{1/2}}}dt.$$

The following two Lemmas are variations of a Lemma by Dunn:

Lemma 7. 

Let $\alpha \in \mathbb{R}\setminus \left\{0\right\}$, $0<\gamma <1$, $N,K,D\in \mathbb{N}$, $c,\ell \in \mathbb{Z}$, r is fixed positive integer, $(r,\ell )=1$ and

$$\mathcal{W}(N,K,D,\gamma )=\sum _{d\le D}\lambda \left(d\right)\sum _{0<\left|k\right|\le K}c\left(k\right)\sum _{\begin{array}{c}n\sim N\\ n\equiv c\left(d\right)\end{array}}{b}_{n}e\left(\alpha k{n}^{\gamma }\right),$$

(16)

where

$$\left|\lambda \left(d\right)\right|\ll 1,c\left(k\right)\ll 1,b_n=\left\{\begin{array}{cc}1,\hfill & \mathit{if}n\mathit{is}\mathit{prime}\mathit{and}n\equiv \ell \left(r\right);\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

(17)

Then

$$\mathcal{W}(N,K,d,\gamma )\ll K^2{N}^{1-\gamma /5+\epsilon }.$$

Proof. 

The proof is the same as that of Lemma 1 in [15]. □

Remark 1. 

We will notice that if in Dunn’s proof we choose

$$M={\displaystyle \frac{N^{1-\frac{3\gamma }{5}}}{K^{\frac{3}{5}}}}$$

then, we obtain the estimate

$$\mathcal{W}(N,K,d,\gamma )\ll K^{4/5}{x}^{1-\gamma /5+\epsilon },$$

which is nontrivial for $\theta <{\displaystyle \frac{\gamma }{4}}$.

Lemma 8. 

Let $\alpha \in \mathbb{R}\setminus \left\{0\right\}$, $b,K,D\in \mathbb{N}$, $x>2$, $c,\ell \in \mathbb{Z}$, r is fixed positive integer, $(r,\ell )=1$, $0<{\alpha }_1<{\alpha }_2<1$, $0<\gamma <{\displaystyle \frac{5}{6}}$ and

$$\mathcal{U}(x,K,D,\gamma )=\sum _{d\le D}\lambda \left(d\right)\sum _{0<\left|k\right|\le K}c\left(k\right)\sum _{x^{{\alpha }_1}<b\le x^{{\alpha }_2}}\eta \left(b\right)\sum _{\begin{array}{c}n\sim x/b\\ bn\equiv c\left(d\right)\\ n\equiv \ell \left(r\right)\end{array}}\rho \left(n\right)e\left(\alpha k{b}^{\gamma }{n}^{\gamma }\right),$$

where $\lambda \left(d\right)\ll 1,c\left(k\right)\ll 1,\eta \left(b\right)\ll 1$ and $\rho \left(n\right)\ll 1$. Then

$$\mathcal{U}(x,K,D,\gamma )\ll K^2{x}^{1-{\displaystyle \frac{\gamma }{5}}+\epsilon }.$$

Proof. 

First, we note that without loss of generality, we may assume that $b\le n$, so $b\le x^{1/2}$ and we will assume that ${\alpha }_2\le 1/2$. Next we rewrite the sum $\mathcal{U}(x,K,D,\gamma )$ on the type

$$\mathcal{U}(x,K,D,\gamma )=\sum _{x^{{\alpha }_1}<b\le x^{{\alpha }_2}}U\left(b\right)$$

(18)

where

$$U\left(b\right)=\sum _{0<\left|k\right|\le K}c\left(k\right)\sum _{\begin{array}{c}n\sim N\\ n\equiv \ell \left(r\right)\end{array}}{a}_n\rho \left(n\right)e\left(\alpha k{b}^{\gamma }{n}^{\gamma }\right),$$

$bN\sim x$, $N\in \mathbb{N}$ and

$$a_n=\sum _{\begin{array}{c}d\le D\\ d|bn-c\end{array}}\lambda \left(d\right)\ll \tau (bn-c){\ll }_{\epsilon }{x}^{\epsilon }$$

Following the proof of Lemma 1 [15] for some parameter $1<M<N$, we estimate the sum $U\left(b\right)$ as

$$U\left(b\right){\ll }_{\epsilon ,r}K{M}^{1/2}{N}^{1/2+\epsilon }+{\displaystyle \frac{K^{1/2}{N}^{3/2-\gamma /2+\epsilon }}{b^{\gamma /2}{M}^{1/2}}}+{\displaystyle \frac{K^2{b}^{\gamma }{M}^2}{N^{1-\gamma }}}.$$

We put

$$M={\displaystyle \frac{N^{1-\frac{3\gamma }{5}}}{b^{\frac{3\gamma }{5}}}}.$$

It follows from the requirement $M>1$, $b\le n$ and $n\sim N$ that $\gamma <5/6$. Having in mind that $bN\sim x$, we get

$$U\left(b\right){\ll }_{\epsilon }{\displaystyle \frac{K{x}^{1-3\gamma /10+\epsilon }}{b}}+{\displaystyle \frac{K^2{x}^{1-\gamma /5+\epsilon }}{b}}$$

Summing over b in (18), we obtain

$$\mathcal{U}(x,K,D,\gamma )\ll K^2{x}^{1-\gamma /5+\epsilon }$$

which is our statement. □

Remark 2. 

We will notice that under the strong condition $\gamma <\frac{5}{6}-\theta$, we can choose

$$M={\displaystyle \frac{N^{1-\frac{3\gamma }{5}}}{K^{\frac{3}{5}}{b}^{\frac{3\gamma }{5}}}}$$

and then we obtain the estimate

$$\mathcal{U}(x,K,D,\gamma )\ll K^{4/5}{x}^{1-\gamma /5+\epsilon },$$

which is nontrivial for $\theta <{\displaystyle \frac{\gamma }{4}}$.

We will also need the following two Lemmas.

Lemma 9. 

Let $0<\Delta <\frac{1}{4},x>2$ and $K={\Delta }^{-1}{log}^{2}x$. Then there exists periodic with period 1 function $\chi \left(t\right)$ such that:

$$\begin{array}{c}0<\chi \left(t\right)\le 1for-\Delta <t<\Delta ,\\ \chi \left(t\right)=0for\Delta \le t\le 1-\Delta ,\end{array}$$

and $\chi \left(t\right)$ admits a Fourier expansion

$$\chi \left(t\right)=\Delta +\Delta \sum _{\left|k\right|>0}g\left(k\right)e\left(kt\right),$$

(19)

where the Fourier coefficients satisfy

$$g\left(k\right)\ll 1forallk\ne 0,\Delta \sum _{\left|k\right|>K}\left|g\left(k\right)\right|\ll x^{-1}.$$

(20)

Proof. 

Such a function is a consequence of Vinogradov’s Lemma (see Chapter 1, [26]). □

Lemma 10. 

Let $0<\Delta <\frac{1}{4},x>2$, $K={\Delta }^{-1}{log}^{2}x$, $D\ge 2$ $\alpha ,\beta ,\gamma \in \mathbb{R}$, $\alpha \ne 0$, $0<\gamma <\frac{5}{6}$, ξ is fixed real number, $0<{\alpha }_1<{\alpha }_2<1$, $c,\ell \in \mathbb{Z}$, r is fixed positive integer, $(r,\ell )=1$, $\lambda \left(d\right)\ll 1,c\left(k\right)\ll 1,\eta \left(b\right)\ll 1$ and $\rho \left(n\right)\ll 1$. Then for the periodic function $\chi \left(t\right)$ from Lemma 9 the equality

$$\begin{array}{cc}\hfill & \sum _{d\le D}\lambda \left(d\right)\sum _{\begin{array}{c}{x}^{{\alpha }_1}\le b<x^{{\alpha }_2}\end{array}}\eta \left(b\right)\sum _{\begin{array}{c}n\sim {\displaystyle \frac{x}{b}}\\ bn\equiv c\left(d\right)\\ n\equiv \ell \left(r\right)\end{array}}\rho \left(n\right)\chi \left(\alpha {(bn+\xi )}^{\gamma }+\beta \right)\hfill \\ \hfill & =\sum _{d\le D}\lambda \left(d\right)\sum _{\begin{array}{c}{x}^{{\alpha }_1}\le b<x^{{\alpha }_2}\end{array}}\eta \left(b\right)\sum _{\begin{array}{c}n\sim {\displaystyle \frac{x}{b}}\\ bn\equiv c\left(d\right)\\ n\equiv \ell \left(r\right)\end{array}}\rho \left(n\right)\chi \left(\alpha {\left(bn\right)}^{\gamma }+\beta \right)+O\left(K^2{x}^{1-\gamma /5+\epsilon }\right)\hfill \end{array}$$

is fulfilled.

Proof. 

From Lemma 9 we get that

$$\begin{array}{cc}\hfill \chi (\alpha {(bn+\xi )}^{\gamma }+\beta )& =\Delta +\Delta \sum _{0<\left|k\right|<K}{g}_1\left(k\right)e\left(\alpha k{(bn+\xi )}^{\gamma }+k\beta \right)+O\left({\displaystyle \frac{1}{x}}\right),\hfill \\ \hfill \chi (\alpha {\left(bn\right)}^{\gamma }+\beta )& =\Delta +\Delta \sum _{0<\left|k\right|<K}{g}_2\left(k\right)e\left(\alpha k{\left(bn\right)}^{\gamma }+k\beta \right)+O\left({\displaystyle \frac{1}{x}}\right),\hfill \end{array}$$

(21)

where $g_1\left(k\right)$ and $g_2\left(k\right)$ satisfy (20). Let $\tilde{g}\left(k\right)=\left(g_2\left(k\right)-g_1\left(k\right)\right)e\left(\beta k\right)$. We have that $\tilde{g}\left(k\right)\ll 1$. From (21) we get

$$\begin{array}{c}\chi (\alpha {(nb+\xi )}^{\gamma }+\beta )-\chi (\alpha {\left(bn\right)}^{\gamma }+\beta )\hfill \\ =-\Delta \sum _{0<\left|k\right|<K}\tilde{g}\left(k\right)e\left(\alpha k{\left(bn\right)}^{\gamma }\right)+O\left(\Delta \sum _{0<\left|k\right|<K}\left|{\tilde{g}}_1(k,b,n)\right|\right),\hfill \end{array}$$

where

$$\begin{array}{c}{\tilde{g}}_1(k,b,n)=g_1\left(k\right)e\left(\beta k\right)\left(e\left(\alpha k{(bn+\xi )}^{\gamma }\right)-e\left(\alpha k{\left(bn\right)}^{\gamma }\right)\right)\hfill \\ \ll {\left|k\right|.|(bn+\xi )}^{\gamma }-{\left(bn\right)}^{\gamma }|.\hfill \end{array}$$

(22)

So

$$\begin{array}{c}\sum _{d\le D}\lambda \left(d\right)\sum _{\begin{array}{c}{x}^{{\alpha }_1}\le b<x^{{\alpha }_2}\end{array}}\eta \left(b\right)\sum _{\begin{array}{c}n\sim {\displaystyle \frac{x}{b}}\\ bn\equiv c\left(d\right)\\ n\equiv \ell \left(r\right)\end{array}}\rho \left(n\right)\left(\chi (\alpha {(nb+\xi )}^{\gamma }+\beta )-\chi (\alpha {\left(bn\right)}^{\gamma }+\beta )\right)\hfill \\ =O\left(R_1\right)+O\left(R_2\right),\hfill \end{array}$$

(23)

where

$$\begin{array}{cc}\hfill R_1& =\Delta \sum _{0<\left|k\right|<K}\tilde{g}\left(k\right)\sum _{d\le D}\lambda \left(d\right)\sum _{\begin{array}{c}{x}^{{\alpha }_1}\le b<x^{{\alpha }_2}\end{array}}\eta \left(b\right)\sum _{\begin{array}{c}n\sim {\displaystyle \frac{x}{b}}\\ bn\equiv c\left(d\right)\\ n\equiv \ell \left(r\right)\end{array}}\rho \left(n\right)e\left(\alpha k{\left(bn\right)}^{\gamma }\right)\hfill \\ \hfill R_2& =\Delta \sum _{0<\left|k\right|<K}\sum _{d\le D}\sum _{\begin{array}{c}{x}^{{\alpha }_1}\le b<x^{{\alpha }_2}\end{array}}\sum _{\begin{array}{c}n\sim {\displaystyle \frac{x}{b}}\\ bn\equiv c\left(d\right)\\ n\equiv \ell \left(r\right)\end{array}}\left|{\tilde{g}}_1(k,b,n)\right|\hfill \end{array}$$

and ${\tilde{g}}_1(k,b,n)$ satisfy (22). To sum $R_1$, we apply Lemma 8 and get

$$R_1\ll K^2{x}^{1-\gamma /5+\epsilon }.$$

(24)

In sum $R_2$ we put $m=bn$ and get

$$R_2\ll \Delta \sum _{0<|k|<K}\sum _{d\le D}\sum _{\begin{array}{c}m\sim x\\ m\equiv c\left(d\right)\end{array}}{\tau \left(m\right)|k|.|(m+\xi )}^{\gamma }-m^{\gamma }|$$

(25)

From $\tau \left(m\right)\ll m^{\epsilon }\ll x^{\epsilon }$ and ${|(m+\xi )}^{\gamma }-m^{\gamma }|\ll x^{\gamma -1}$ we receive

$$\begin{array}{cc}\hfill R_2& \ll \Delta x^{\gamma -1+\epsilon }\sum _{0<k<K}k\sum _{d\le D}\sum _{\begin{array}{c}m\sim x\\ m\equiv c\left(d\right)\end{array}}1\hfill \\ \hfill & \ll \Delta K^2{x}^{\gamma +\epsilon }.\hfill \end{array}$$

(26)

From (23), (24), and (26), we obtain our statement. □

Remark 3. 

We notice that if we use result in Remark 2 we get that

$$\begin{array}{cc}\hfill & \sum _{d\le D}\lambda \left(d\right)\sum _{\begin{array}{c}{x}^{{\alpha }_1}\le b<x^{{\alpha }_2}\end{array}}\eta \left(b\right)\sum _{\begin{array}{c}n\sim {\displaystyle \frac{x}{b}}\\ bn\equiv c\left(d\right)\\ n\equiv \ell \left(r\right)\end{array}}\rho \left(n\right)\chi \left(\alpha {(bn+\xi )}^{\gamma }+\beta \right)\hfill \\ \hfill & =\sum _{d\le D}\lambda \left(d\right)\sum _{\begin{array}{c}{x}^{{\alpha }_1}\le b<x^{{\alpha }_2}\end{array}}\eta \left(b\right)\sum _{\begin{array}{c}n\sim {\displaystyle \frac{x}{b}}\\ bn\equiv c\left(d\right)\\ n\equiv \ell \left(r\right)\end{array}}\rho \left(n\right)\chi \left(\alpha {\left(bn\right)}^{\gamma }+\beta \right)+O\left(K^{4/5}{x}^{1-\gamma /5+\epsilon }\right),\hfill \end{array}$$

whit γ and θ satisfying (4).

4. Proof of the Theorem 1

We will prove that there are infinitely many primes p such that $\left|\right|\alpha p^{\gamma }+\beta \left|\right|<p^{-\theta }$ and $p+2=P_2$.

First, using the ideas of Chen (see [11]), we consider the sum

$$S=\sum _{\begin{array}{c}p\sim x\\ (p+2,P(z\left)\right)=1\end{array}}\chi (\alpha p^{\gamma }+\beta )(1-{\displaystyle \frac{1}{2}}\sum _{\begin{array}{c}z\le p_1<x^{1/3}\\ p_1\mid p+2\end{array}}1-{\displaystyle \frac{1}{2}}\sum _{\begin{array}{c}z\le p_1<x^{1/3}\\ x^{1/3}\le p_2<{(x/p_1)}^{1/2}\\ p_3\ge p_2\\ p+2=p_1{p}_2{p}_3\end{array}}1).$$

(27)

Here, $\chi$ is a periodic function as described in Lemma 9. The variables z, $\Delta$, and K are defined according to (5). We also have the following conditions:

$$0<\theta <{\displaystyle \frac{\gamma }{10}},8<\delta <10,$$

(28)

where $\delta$ will be chosen later in the discussion.

We will notice that the expression in brackets is

• 1 if $p+2$ has no prime divisors smaller than $x^{1/3}$;
• ${\displaystyle \frac{1}{2}}$ if $p+2$ has exactly one prime divisor in the interval $[z,x^{1/3})$ and one prime divisor greater than $x^{1/3}$;
• ≤0 in all other cases.

So, the weight of p is positive only if

$$p+2=P_2\mathrm{and}\left|\right|\alpha p^{\gamma }+\beta \left|\right|<\Delta .$$

Therefore, it is sufficient to demonstrate that $S>0$.

Our next step is to express S in the form (see (27)):

$$S=S_1-{\displaystyle \frac{1}{2}}{S}_2-{\displaystyle \frac{1}{2}}{S}_3,$$

(29)

where

$$\begin{array}{cc}\hfill S_1=& \sum _{\begin{array}{c}p\sim x\\ (p+2,P(z\left)\right)=1\end{array}}\chi (\alpha p^{\gamma }+\beta ),\hfill \\ \hfill S_2=& \sum _{\begin{array}{c}z\le p_1<x^{1/3}\end{array}}\sum _{\begin{array}{c}p\sim x\\ (p+2,P(z\left)\right)=1\\ p\equiv -2\left(p_1\right)\end{array}}\chi (\alpha p^{\gamma }+\beta ),\hfill \\ \hfill S_3=& \sum _{\begin{array}{c}z\le p_1<x^{1/3}\\ x^{1/3}\le p_2<{({\displaystyle \frac{x}{p_1}})}^{{\displaystyle \frac{1}{2}}}\\ p_3\ge p_2\end{array}}\sum _{\begin{array}{c}p\sim x\\ p+2=p_1{p}_2{p}_3\end{array}}\chi (\alpha p^{\gamma }+\beta ).\hfill \end{array}$$

We will estimate the sums $S_1\left(x\right)$, $S_2\left(x\right)$, and $S_3\left(x\right)$ separately, the first one from below and the other two from above.

4.1. Estimate of Sum $S_1$

To sum $S_1$, we apply the lower bound linear sieve in the form due to Iwaniec (see Lemma 2). Let ${\lambda }^{-}$ be a lower bound well factorable function of level $D_1$ with

$$D_1=x^{4/7-\epsilon },z=x^{1/\delta },\mathrm{and}{s}_1={\displaystyle \frac{log{D}_1}{logz}}$$

(30)

and

$$\omega \left(d\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}2|d,\hfill \\ {\displaystyle \frac{d}{\phi \left(d\right)}},\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

From the Fourier expansion (19) of the function $\chi (\alpha p^{\gamma }+\beta )$ and the inequalities (20), we obtain:

$$S_1\ge \Delta \left(S_1^{\prime }+E_1+R_1\right)+O\left(1\right),$$

(31)

where

$$\begin{array}{cc}\hfill S_1^{\prime }=& \mathrm{Li}\left(x\right)V\left(z\right)\left(f\left(s_1\right)-o\left(1\right)\right),\hfill \\ \hfill R_1=& \sum _{\begin{array}{c}d\le x^{4/7-\epsilon }\\ d\left|P\right(z)\end{array}}{\lambda }^{-}\left(d\right)(\sum _{\begin{array}{c}p\sim x\\ p\equiv -2\left(d\right)\end{array}}1-{\displaystyle \frac{Lix}{\phi \left(d\right)}}),\hfill \\ \hfill E_1=& \sum _{\begin{array}{c}d\le x^{4/7-\epsilon }\\ d\left|P\right(z)\end{array}}{\lambda }^{-}\left(d\right)\sum _{0<\left|k\right|<K}{c}_k\sum _{\begin{array}{c}p\sim x\\ p\equiv -2\left(d\right)\end{array}}e\left(\alpha k{p}^{\gamma }\right)\hfill \end{array}$$

(32)

and

$$\begin{array}{cc}\hfill V\left(z\right)& =\prod _{2<p<z}\left(1-{\displaystyle \frac{1}{p-1}}\right)={\displaystyle \frac{2C_2{e}^{-\gamma }}{logz}}\left(1+O\left({\displaystyle \frac{1}{logz}}\right)\right),\hfill \\ \hfill C_2& =\prod _{p\ge 3}\left(1-{\displaystyle \frac{1}{{(p-1)}^2}}\right)\approx 0.660161815846869\dots \hfill \end{array}$$

(33)

As

$$\begin{array}{cc}\hfill R_1=& \sum _{\begin{array}{c}d\le x^{4/7-\epsilon }\end{array}}{\lambda }^{-}\left(d\right)\left(\sum _{\begin{array}{c}p\le 2x\\ p\equiv -2\left(d\right)\end{array}}1-{\displaystyle \frac{\mathrm{Li}\left(2x\right)}{\phi \left(d\right)}}\right)-\sum _{\begin{array}{c}d\le x^{4/7-\epsilon }\end{array}}{\lambda }^{-}\left(d\right)\left(\sum _{\begin{array}{c}p\le x\\ p\equiv 1\left(d\right)\end{array}}1-{\displaystyle \frac{\mathrm{Li}\left(x\right)}{4\phi \left(d\right)}}\right)\hfill \end{array}$$

from Theorem 5, we get

$$R_1\ll {\displaystyle \frac{x}{{(logx)}^A}}.$$

(34)

Applying Lemma 7 with $r=1$ and using (28), we obtain

$$E_1\ll K^2{N}^{1-\gamma /5+\epsilon }\ll {\displaystyle \frac{x}{{(logx)}^A}}.$$

(35)

Using (31), (32), (34), (35), and Lemma 3, we get

$$\begin{array}{c}{S}_1\ge {\displaystyle \frac{2e^{\gamma }\Delta Li\left(x\right)V\left(z\right)}{s_1}}(log(s_1-1)\hfill \\ +{\int }_3^{s_1-1}{\displaystyle \frac{dt}{t}}{\int }_2^{t-1}{\displaystyle \frac{log(u-1)}{u}}du)+O\left({\displaystyle \frac{x}{{log}^{3}x}}\right).\hfill \end{array}$$

(36)

4.2. Estimate of Sum $S_2$

To the inner sum in $S_2$, we apply upper bound linear sieve (see Lemma 1) with

$$D_2={\displaystyle \frac{x^{1/2-\epsilon }}{p_1}}\mathrm{and}z=x^{1/\delta }.$$

Let ${\lambda }^{+}\left(d\right)$ be upper bounds of Rosser’s weights of level $D_2$. Using the Fourier expansion (19) of function $\chi (\alpha p^{\gamma }+\beta )$ and the inequalities (20), we obtain

$$S_2\le \Delta \left(S_2^{\prime }+E_2+R_2\right)+O\left(1\right),$$

(37)

where

$$\begin{array}{cc}\hfill S_2^{\prime }=& \mathrm{Li}\left(\mathrm{x}\right)\sum _{\begin{array}{c}z\le p_1<x^{1/3}\end{array}}{\displaystyle \frac{1}{p_1-1}}\sum _{\begin{array}{c}d\le D_2\\ d\left|P\right(z)\end{array}}{\displaystyle \frac{{\lambda }^{+}\left(d\right)}{\phi \left(d\right)}}\hfill \\ \hfill R_2=& \sum _{\begin{array}{c}z\le p_1<x^{1/3}\end{array}}\sum _{\begin{array}{c}d\le D_2\\ d\left|P\right(z)\end{array}}{\lambda }^{+}\left(d\right)\left(\sum _{\begin{array}{c}p\sim x\\ p\equiv -2\left(p_{1}d\right)\end{array}}1-{\displaystyle \frac{Lix}{\phi \left(d\right)}}\right)\hfill \\ \hfill E_2=& \sum _{0<\left|k\right|<K}{c}_k\sum _{\begin{array}{c}z\le p_1<x^{1/3}\end{array}}\sum _{\begin{array}{c}d\le D_2\\ d\left|P\right(z)\end{array}}{\lambda }^{+}\left(d\right)\sum _{\begin{array}{c}p\sim x\\ p\equiv -2\left(p_{1}d\right)\end{array}}e\left(\alpha k{p}^{\gamma }\right)\hfill \end{array}$$

We can rewrite the sum $R_2$ in the type

$$R_2=\sum _{q\le x^{1/2-\epsilon }}\upsilon \left(q\right)\left(\sum _{\begin{array}{c}p\sim x\\ p\equiv -2\left(q\right)\end{array}}1-{\displaystyle \frac{Lix}{\phi \left(q\right)}}\right),$$

(38)

where

$$\upsilon \left(q\right)=\left\{\begin{array}{cc}{\lambda }^{+}\left(d\right),\hfill & \mathrm{if}q=d{p}_1,d|P\left(z_2\right),z_2\le p_1<x^{1/3},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(39)

from

$$\begin{array}{cc}\hfill R_2\le & \sum _{\begin{array}{c}q\le x^{1/2-\epsilon }\end{array}}\nu \left(q\right)|\sum _{\begin{array}{c}p\le 2x\\ p\equiv -2\left(d\right)\end{array}}1-{\displaystyle \frac{\mathrm{Li}\left(2x\right)}{\phi \left(d\right)}}|+\sum _{\begin{array}{c}q\le x^{1/2-\epsilon }\end{array}}\nu \left(q\right)|\sum _{\begin{array}{c}p\le x\\ p\equiv 1\left(d\right)\end{array}}1-{\displaystyle \frac{\mathrm{Li}\left(x\right)}{4\phi \left(d\right)}}|\hfill \end{array}$$

and from Bombieri–Vinogradov’s theorem (see Theorem 4), we get

$$R_2\ll {\displaystyle \frac{x}{{(logx)}^A}}.$$

Using the notation (39), Lemma 7 with $r=1$, and restriction (5), we get

$$\begin{array}{cc}\hfill E_2=& \sum _{0<\left|k\right|<K}{c}_k\sum _{q\le x^{1/2-\epsilon }}\upsilon \left(q\right)\sum _{\begin{array}{c}p\sim x\\ p\equiv -2\left(q\right)\end{array}}e\left(\alpha k{p}^{\gamma }\right)\hfill \\ \hfill \ll & K^2{x}^{1-\gamma /5+\epsilon }\hfill \\ \hfill \ll & {\displaystyle \frac{x}{{(logx)}^A}}.\hfill \end{array}$$

(40)

Let

$$s_2={\displaystyle \frac{\delta }{2}}-\epsilon \delta .$$

(41)

First, we apply linear sieve (Lemma 1) to sum $S_2^{\prime }$:

$$S_2^{\prime }\le \mathrm{Li}\left(\mathrm{x}\right)V\left(z\right)\sum _{z\le p<x^{1/3}}{\displaystyle \frac{1}{p}}F\left(s_2-{\displaystyle \frac{\delta logp}{logx}}\right)+O\left({\displaystyle \frac{x}{{log}^{3}x}}\right).$$

and then the Abel transformation:

$$\begin{array}{cc}\hfill S_2^{\prime }\le & \mathrm{Li}\left(\mathrm{x}\right)V\left(z\right)\underset{z}{\overset{x^{1/3}}{\int }}{\displaystyle \frac{1}{tlogt}}F\left(s_2-{\displaystyle \frac{\delta logt}{logx}}\right)dt+O\left({\displaystyle \frac{x}{{log}^{3}x}}\right)\hfill \\ \hfill \le & \mathrm{Li}\left(\mathrm{x}\right)V\left(z\right)\underset{1/\delta }{\overset{1/3}{\int }}{\displaystyle \frac{F(s_2-\delta u)}{u}}du+O\left({\displaystyle \frac{x}{{log}^{3}x}}\right).\hfill \end{array}$$

Finally, from Lemma 3 and the inequalities (37), (38), and (40), we get

$$\begin{array}{c}{S}_2\le {\displaystyle \frac{2e^{\gamma }\Delta \mathrm{Li}\left(\mathrm{x}\right)V\left(z\right)}{s_2}}(log{\displaystyle \frac{s_2-1}{1/2-3\epsilon }}+log\left(\delta s_2-\delta \right)\underset{2}{\overset{s_2-2}{\int }}{\displaystyle \frac{log(u-1)}{u}}du\hfill \\ \hfill -\delta \underset{1/\delta }{\overset{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{3}{\delta }}-\epsilon }{\int }}{\displaystyle \frac{\left(log\left(s_2-\delta u\right)-logu\right)log\left(s_2-\delta u-2\right)}{s_2-\delta u-1}}du)+O\left({\displaystyle \frac{x}{{log}^{3}x}}\right).\end{array}$$

(42)

4.3. Estimate of Sum $S_3$

It is not difficult to see that

$$S_3\le \sum _{\begin{array}{c}z\le p_1<x^{1/3}\end{array}}\sum _{\begin{array}{c}{x}^{1/3}\le p_2<{(x/p_1)}^{1/2}\end{array}}\sum _{\begin{array}{c}{p}_3\sim {\displaystyle \frac{x}{p_1{p}_2}}\\ (p_1{p}_2{p}_3-2,P\left(z\right))=1\end{array}}\chi (\alpha {(p_1{p}_2{p}_3-2)}^{\gamma }+\beta ).$$

Next, we apply the upper bound linear sieve (see Lemma 1) with ${\lambda }^{+}\left(d\right)$—upper bounds Rosser’s weights of level $D_2=x^{1/2-\epsilon }$ and $z=x^{1/\delta }$:

$$\begin{array}{cc}\hfill S_3& \le \sum _{d\le D_2}{\lambda }^{+}\left(d\right)\sum _{\begin{array}{c}z\le p_1<x^{1/3}\end{array}}\sum _{\begin{array}{c}{x}^{1/3}\le p_2<{(x/p_1)}^{1/2}\end{array}}\sum _{\begin{array}{c}{p}_3\sim {\displaystyle \frac{x}{p_1{p}_2}}\\ p_1{p}_2{p}_3\equiv 2\left(d\right)\end{array}}\chi (\alpha {(p_1{p}_2{p}_3-2)}^{\gamma }+\beta )\hfill \\ \hfill & \le \sum _{d\le D_2}{\lambda }^{+}\left(d\right)\sum _{\begin{array}{c}z{x}^{1/3}\le b<x^{2/3}\end{array}}\eta \left(b\right)\sum _{\begin{array}{c}{p}_3\sim {\displaystyle \frac{x}{b}}\\ b{p}_3\equiv 2\left(d\right)\end{array}}\chi (\alpha {(b{p}_3-2)}^{\gamma }+\beta ),\hfill \end{array}$$

where

$$\eta \left(b\right)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}b=p_1{p}_2,z\le p_1<x^{1/3},x^{1/3}\le p_2<{(x/p_1)}^{1/2};\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(43)

To the last sum, we apply Lemma 10 and get

$$S_3\le {\tilde{S}}_3+O\left(K^2{x}^{1-\gamma /5+\epsilon }\right),$$

(44)

where

$${\tilde{S}}_3=\sum _{d\le D_2}{\lambda }^{+}\left(d\right)\sum _{\begin{array}{c}z{x}^{1/3}\le b<x^{2/3}\end{array}}h\left(b\right)\sum _{\begin{array}{c}{p}_3\sim {\displaystyle \frac{x}{b}}\\ b{p}_3\equiv 2\left(d\right)\end{array}}\chi (\alpha {\left(b{p}_3\right)}^{\gamma }+\beta )$$

(45)

Next, using Fourier expansion (19) of function $\chi$, the inequalities (20), and (45), we obtain

$${\tilde{S}}_3\le \Delta \left(S_3^{\prime }+E_3+R_3\right),$$

(46)

where

$$\begin{array}{cc}\hfill S_3^{\prime }\le & \sum _{\begin{array}{c}d\le x^{1/2-\epsilon }\\ d\left|P\right(z)\end{array}}{\displaystyle \frac{{\lambda }^{+}\left(d\right)}{\phi \left(d\right)}}\sum _{\begin{array}{c}z\le p_1<x^{1/3}\end{array}}\sum _{\begin{array}{c}{x}^{1/3}\le p_2<{(x/p_1)}^{1/2}\end{array}}\mathrm{L}i\left({\displaystyle \frac{x}{p_1{p}_2}}\right),\hfill \\ \hfill R_3=& \sum _{\begin{array}{c}d\le x^{1/2-\epsilon }\\ d\left|P\right(z)\end{array}}{\lambda }^{+}\left(d\right)\sum _{\begin{array}{c}z{x}^{1/3}\le b<x^{2/3}\end{array}}\eta \left(b\right)(\sum _{\begin{array}{c}{p}_3\sim {\displaystyle \frac{x}{b}}\\ b{p}_3\equiv 2\left(d\right)\end{array}}1-{\displaystyle \frac{Li(x/b)}{\phi \left(d\right)}}),\hfill \\ \hfill E_3=& \sum _{\begin{array}{c}z{x}^{1/3}\le b<x^{2/3}\end{array}}\eta \left(b\right)\sum _{0<\left|k\right|<K}{c}_k\sum _{\begin{array}{c}d\le x\\ d\left|P\right(z)\end{array}}\lambda \left(d\right)\sum _{\begin{array}{c}{p}_3\sim {\displaystyle \frac{x}{b}}\\ b{p}_3\equiv 2\left(d\right)\end{array}}e\left(\alpha b^{\gamma }k{p}_3^{\gamma }\right).\hfill \end{array}$$

It is obvious that

$$\begin{array}{cc}\hfill R_3\le & \sum _{\begin{array}{c}d\le x^{1/2-\epsilon }\\ d\left|P\right(z)\end{array}}\underset{amodd}{max}|\sum _{\begin{array}{c}z{x}^{1/3}\le b<x^{2/3}\end{array}}\eta \left(b\right)(\sum _{\begin{array}{c}{p}_3\le {\displaystyle \frac{2x}{b}}\\ b{p}_3\equiv a\left(d\right)\end{array}}1-{\displaystyle \frac{Li(x/b)}{\phi \left(d\right)}})|\hfill \\ \hfill +& \sum _{\begin{array}{c}d\le x^{1/2-\epsilon }\\ d\left|P\right(z)\end{array}}\underset{amodd}{max}|\sum _{\begin{array}{c}z{x}^{1/3}\le b<x^{2/3}\end{array}}\eta \left(b\right)(\sum _{\begin{array}{c}{p}_3\le {\displaystyle \frac{x}{b}}\\ b{p}_3\equiv a\left(d\right)\end{array}}1-{\displaystyle \frac{Li(x/b)}{\phi \left(d\right)}})|\hfill \end{array}$$

(47)

Then, from Theorem 6, we get

$$R_3\ll {\displaystyle \frac{x}{{(logx)}^A}}.$$

(48)

To estimate the sum $E_3$, we apply Lemma 8 with $r=1$, and from the restriction (5), we get

$$E_3\ll K^2{x}^{1-\gamma /5+\epsilon }\ll {\displaystyle \frac{x}{{(logx)}^A}}.$$

(49)

To sum $S_3^{\prime }$, we apply Lemma 1 with $s_2$ satisfying (41):

$$\begin{array}{c}{S}_3^{\prime }\le xV\left(z\right)F\left(s_2\right)\sum _{\begin{array}{c}z\le p_1<x^{1/3}\end{array}}{\displaystyle \frac{1}{p_1}}\sum _{\begin{array}{c}{x}^{{\displaystyle \frac{1}{3}}}\le p_2<{(x/p_1)}^{{\displaystyle \frac{1}{2}}}\end{array}}{\displaystyle \frac{1}{p_2(logx-log{p}_1-log{p}_2)}}\hfill \\ +O\left({\displaystyle \frac{x}{{log}^{3}x}}\right).\hfill \end{array}$$

Then, by applying Abel’s identity twice, we obtain:

$$\begin{array}{cc}\hfill S_3^{\prime }\le & {\displaystyle \frac{xV\left(z\right)F\left(s_2\right)}{logx}}\sum _{\begin{array}{c}z\le p_1<x^{1/3}\end{array}}{\displaystyle \frac{1}{p_1}}\underset{1/3}{\overset{{\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{log{p}_1}{logx}}\right)}{\int }}{\displaystyle \frac{1}{u(1-\frac{log{p}_1}{logx}-u)}}du+O\left({\displaystyle \frac{x}{{log}^{3}x}}\right)\hfill \\ \hfill =& {\displaystyle \frac{xV\left(z\right)F\left(s_2\right)}{logx}}\underset{1/\delta }{\overset{1/3}{\int }}{\displaystyle \frac{log(2-3u)}{u(1-u)}}du+O\left({\displaystyle \frac{x}{{log}^{3}x}}\right).\hfill \end{array}$$

(50)

From inequalities (46), (48)–(50) follows

$$S_3\le {\displaystyle \frac{\Delta xV\left(z\right)F\left(s_2\right)}{logx}}\underset{1/\delta }{\overset{1/3}{\int }}{\displaystyle \frac{log(2-3u)}{u(1-u)}}du+O\left({\displaystyle \frac{x}{{log}^{3}x}}\right).$$

(51)

4.4. Estimate of Sum S

Now from (29), (36), (42), (51) we get

$$\begin{array}{cc}\hfill S& \ge {\displaystyle \frac{\Delta xV\left(z\right)}{logx}}({\displaystyle \frac{2e^{\gamma }log(s_1-1)}{s_1}}+{\displaystyle \frac{2e^{\gamma }}{s_1}}{\int }_3^{s_1-1}{\displaystyle \frac{dt}{t}}{\int }_2^{t-1}{\displaystyle \frac{log(u-1)}{u}}du\hfill \\ \hfill & -{\displaystyle \frac{2e^{\gamma }}{s_2}}log{\displaystyle \frac{s_2-1}{1/2-3\epsilon }}-{\displaystyle \frac{2e^{\gamma }log\left(\delta s_2-\delta \right)}{s_2}}\underset{2}{\overset{s_2-2}{\int }}{\displaystyle \frac{log(u-1)}{u}}du\hfill \\ \hfill & +{\displaystyle \frac{2e^{\gamma }\delta }{s_2}}\underset{1/\delta }{\overset{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{3}{\delta }}-\epsilon }{\int }}{\displaystyle \frac{\left(log(s_2-\delta u)-logu\right)log(s_2-\delta u-2)}{s_2-\delta u-1}}du\hfill \\ \hfill & -F\left(s_2\right)\underset{1/\delta }{\overset{1/3}{\int }}{\displaystyle \frac{log(2-3u)}{u(1-u)}}du)\hfill \end{array}$$

So for $\delta =9$ and $\epsilon =0.0001$ the inequality

$$S\ge {\displaystyle \frac{c\Delta x}{{log}^{2}x}}$$

(52)

is fulfilled for a small $c>0$. Theorem 1 is proved.

If, in the estimates of $E_2$ and $E_3$, we use Remark 2 and Remark 3, respectively, we will obtain for the sum S the same lower bound as in (52), but under the restrictions (4), and obtain Theorem 2.

5. Proof of the Theorem 3

In this section, we will prove that there are infinitely many primes p of the form $p=a^2+b^2+1$ and such that $\left|\right|\alpha p^{\gamma }+\beta \left|\right|<p^{-\theta }$.

We will utilize the fact that if $p-1=a^2+b^2$ and $(a,b)=1$, then $p-1$ has no prime factors belonging to $P_3=\left\{p\right|p\equiv 3(mod4)\}$. Our purpose will be an evaluation of the sum

$$\mathbb{S}\left(x\right)=\sum _{\begin{array}{c}p\sim x\\ \left|\right|\alpha p^{\gamma }+\beta \left|\right|<\Delta \\ (p-1,{\mathcal{P}}_3)=1\end{array}}1.$$

From here to the end of the paragraph, we will use the notations (12). It is not difficult to see that

$$\mathbb{S}\left(x\right)\ge \sum _{\begin{array}{c}p\sim x\\ p\equiv 3\left(8\right)\\ (p-1,{\mathcal{P}}_3\left(z\right))=1\end{array}}\chi (\alpha p^{\gamma }+\beta )\left(1-A(p,z,2x)\right)={\mathbb{S}}_1\left(x\right)-{\mathbb{S}}_2\left(x\right),$$

where

$$A(p,z,2x)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\mathrm{there}\mathrm{exist}{p}_1\in P_3,z\le p_1\le 2x,p_1|p-1;\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(53)

and

$$\begin{array}{cc}\hfill {\mathbb{S}}_1\left(x\right)=& \sum _{\begin{array}{c}p\sim x\\ p\equiv 3\left(8\right)\\ (p-1,{\mathcal{P}}_3\left(z\right))=1\end{array}}\chi (\alpha p^{\gamma }+\beta ),\hfill \\ \hfill {\mathbb{S}}_2\left(x\right)=& \sum _{\begin{array}{c}p\sim x\\ p\equiv 3\left(8\right)\\ (p-1,{\mathcal{P}}_3\left(z\right))=1\end{array}}A(p,z,2x)\chi (\alpha p^{\gamma }+\beta ).\hfill \end{array}$$

To prove our assertion, it is sufficient to verify that ${\mathbb{S}}_1\left(x\right)-{\mathbb{S}}_2\left(x\right)>0$. We will estimate the sums ${\mathbb{S}}_1\left(x\right)$ and ${\mathbb{S}}_2\left(x\right)$ separately, from below and above, respectively.

5.1. Estimation of Sum ${\mathbb{S}}_1\left(x\right)$

To sum ${\mathbb{S}}_1\left(x\right),$ we apply a half-dimensional sieve. Let ${\lambda }_{sem}^{-}$ be lower bound weights of half dimensional sieve of order $D=x^{1/2-\epsilon }$. Then

$${\mathbb{S}}_1\left(x\right)\ge \sum _{\begin{array}{c}d\le D\\ d|P_3\left(z\right)\end{array}}{\lambda }_{sem}^{-}\left(d\right)\sum _{\begin{array}{c}p\sim x\\ p\equiv 1\left(d\right)\\ p\equiv 3\left(8\right)\end{array}}\chi (\alpha p^{\gamma }+\beta )$$

Applying the Fourier expansion (19) and the inequalities (20), we obtain:

$${\mathbb{S}}_1\ge \Delta \left(S_{sem}^{\prime }+E_{sem}+R_{sem}\right)+O\left(1\right),$$

(54)

where

$$\begin{array}{cc}\hfill S_{sem}^{\prime }=& {\displaystyle \frac{\mathrm{Li}\left(2x\right)-\mathrm{Li}\left(x\right)}{4}}\sum _{\begin{array}{c}d\le D\\ d|P_3\left(z\right)\end{array}}{\displaystyle \frac{{\lambda }_{sem}^{-}\left(d\right)}{\phi \left(d\right)}},\hfill \\ \hfill R_{sem}=& \sum _{\begin{array}{c}d\le D\\ d|P_3\left(z\right)\end{array}}{\lambda }_{sem}^{-}\left(d\right)(\sum _{\begin{array}{c}p\sim x\\ p\equiv 1\left(d\right)\\ p\equiv 3\left(8\right)\end{array}}1-{\displaystyle \frac{\mathrm{Li}\left(2x\right)-\mathrm{Li}\left(x\right)}{4\phi \left(d\right)}}),\hfill \\ \hfill E_{sem}=& \sum _{\begin{array}{c}d\le D\\ d|P_3\left(z\right)\end{array}}{\lambda }_{sem}^{-}\left(d\right)\sum _{0<\left|k\right|\le K}c\left(k\right)\sum _{\begin{array}{c}p\sim x\\ p\equiv 1\left(d\right)\\ p\equiv 3\left(8\right)\end{array}}e\left(\alpha k{p}^{\gamma }\right).\hfill \end{array}$$

It is clear that $S_{sem}^{\prime }=S(\mathcal{A},P_3,z)$ (see (12)). Then, from Lemma 5, we get

$$S_{sem}^{\prime }\ge {\displaystyle \frac{A{C}_{3}x}{\sqrt{2}{(logx)}^{3/2}}}\underset{1}{\overset{\delta /2}{\int }}{\displaystyle \frac{dt}{\sqrt{t(t-1)}}}+o\left({\displaystyle \frac{x}{{(logx)}^{3/2}}}\right)$$

(55)

with A and $C_3$ consequently defined by (13) and (14).

As

$$\begin{array}{cc}\hfill R_{sem}\ll & \sum _{\begin{array}{c}d\le D\end{array}}|\sum _{\begin{array}{c}p\le x\\ p\equiv 1\left(d\right)\\ p\equiv 3\left(8\right)\end{array}}1-{\displaystyle \frac{\mathrm{Li}\left(x\right)}{4\phi \left(d\right)}}|+\sum _{\begin{array}{c}d\le D\end{array}}|\sum _{\begin{array}{c}p\le 2x\\ p\equiv 1\left(d\right)\\ p\equiv 3\left(8\right)\end{array}}1-{\displaystyle \frac{\mathrm{Li}\left(2x\right)}{4\phi \left(d\right)}}|\hfill \\ \hfill \ll & \sum _{\begin{array}{c}d\le 8D\end{array}}\underset{amod8d}{max}|\sum _{\begin{array}{c}p\le x\\ p\equiv a\left(8d\right)\end{array}}1-{\displaystyle \frac{\mathrm{Li}\left(x\right)}{4\phi \left(d\right)}}|+\sum _{\begin{array}{c}d\le 8D\end{array}}\underset{amod8d}{max}|\sum _{\begin{array}{c}p\le 2x\\ p\equiv a\left(8d\right)\end{array}}1-{\displaystyle \frac{\mathrm{Li}\left(2x\right)}{4\phi \left(d\right)}}|\hfill \end{array}$$

from Bombieri–Vinogradov’s Theorem (Theorem 4), it follows that

$$R_{sem}\ll {\displaystyle \frac{x}{{(logx)}^A}}.$$

(56)

To estimate the sum $E_{sem}\left(x\right)$, we apply Lemma 7 for $r=8$. Using (5) we obtain

$$E_{sem}\ll {\displaystyle \frac{x}{{(logx)}^A}}.$$

(57)

From (54)–(57) it follows that

$${\mathbb{S}}_1\left(x\right)\ge {\displaystyle \frac{A{C}_3\Delta x}{\sqrt{2}{(logx)}^{3/2}}}\underset{1}{\overset{\delta /2}{\int }}{\displaystyle \frac{dt}{\sqrt{t(t-1)}}}+o\left({\displaystyle \frac{x}{{(logx)}^{3/2}}}\right)$$

(58)

5.2. Estimation of Sum ${\mathbb{S}}_2$

Having in mind the definition (53) of $A(p,z,2x)$, we observe that ${\mathbb{S}}_2\left(x\right)$ is the sum over that primes p such that $p-1$ has prime divisor of set $[z,2x]\cap P_3$. As each element $p-1\in \mathcal{A}$ is divisible by an even number of primes from ${\mathcal{P}}_3$ and $2\Vert p-1$, we conclude that ${\mathbb{S}}_2\left(x\right)$ is the sum over prime numbers of the form $p=1+2n{p}_1{p}_2$ whith $p_1,p_2\in {\mathcal{P}}_3$, $p_2\ge p_1\ge x^{1/\delta }$, and n is an integer divisible only by primes of the form $p\equiv 1(mod4)$. So

$${\mathbb{S}}_2\left(x\right)=\sum _{\begin{array}{c}x<p<2x\\ p=1+2n{p}_1{p}_2\\ p_1\equiv p_2\equiv 3\left(4\right)\\ q|n\Rightarrow q\equiv 1(4)\end{array}}\chi (\alpha p^{\gamma }+\beta )$$

From $p=1+2n{p}_1{p}_2\le 2x$ follows $z\le p_1\le {\left(\frac{x}{n}\right)}^{1/2}$, $n\le {\displaystyle \frac{x}{z^2}}$. So

$${\mathbb{S}}_2\left(x\right)\le \sum _{\begin{array}{c}n\le {\displaystyle \frac{x}{z^2}}\\ q|n\Rightarrow q\equiv 1(4)\end{array}}\sum _{\begin{array}{c}z\le p_1\le {\left(\frac{x}{n}\right)}^{1/2}\\ p_1\equiv 3\left(4\right)\end{array}}\sum _{\begin{array}{c}{p}_2\sim {\displaystyle \frac{x}{2n{p}_1}}\\ p_2\equiv 3\left(4\right)\\ (1+2n{p}_1{p}_2,P\left(z\right))=1\end{array}}\chi (\alpha {(1+2n{p}_1{p}_2)}^{\gamma }+\beta ).$$

Let $b=n{p}_1$ and let

$$\eta \left(b\right)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}b=n{p}_1,z\le p_1\le {\left({\displaystyle \frac{x}{n}}\right)}^{1/2},p_1\equiv 3\left(4\right),(n,P_3\left(z\right))=1\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

Then $z\le b\le x/z$ and

$${\mathbb{S}}_2\left(x\right)\le \sum _{z<b<x/z}\eta \left(b\right)\sum _{\begin{array}{c}{p}_2\sim {\displaystyle \frac{x}{2b}}\\ (1+2b{p}_2,P\left(z\right))=1\\ p_2\equiv 3\left(4\right)\end{array}}\chi (\alpha {(2b{p}_2+1)}^{\gamma }+\beta ).$$

To the sum over $p_2$ in the above inequality, we apply the upper bound linear sieve (see Lemma 1) with ${\lambda }^{+}\left(d\right)$—upper bounds Rosser’s weights of level $D=x^{1/2-\epsilon }$ and $z=x^{1/\delta }$:

$${\mathbb{S}}_2\left(x\right)\le \sum _{d\le D}{\lambda }^{+}\left(d\right)\sum _{z<b<x/z}\eta \left(b\right)\sum _{\begin{array}{c}{p}_2\sim {\displaystyle \frac{x}{2b}}\\ 2b{p}_2\equiv -1\left(d\right)\\ p_2\equiv 3\left(4\right)\end{array}}\chi (\alpha {(2b{p}_2+1)}^{\gamma }+\beta ).$$

Then, with the help of Lemma 10, we get

$${\mathbb{S}}_2\left(x\right)\le {\mathbb{S}}_2^{\prime }\left(x\right)+O\left(K^2{x}^{1-\gamma /5+\epsilon }\right),$$

(59)

where

$${\mathbb{S}}_2^{\prime }\left(x\right)=\sum _{d\le D_2}{\lambda }^{+}\left(d\right)\sum _{z<b<x/z}\eta \left(b\right)\sum _{\begin{array}{c}{p}_2\sim {\displaystyle \frac{x}{2b}}\\ 2b{p}_2\equiv -1\left(d\right)\\ p_2\equiv 3\left(4\right)\end{array}}\chi (\alpha {\left(2b{p}_2\right)}^{\gamma }+\beta )$$

To sum ${\mathbb{S}}_2^{\prime }\left(x\right)$, we apply the Fourier expansion (19) and the inequalities (20):

$${\mathbb{S}}_2^{\prime }\left(x\right)\le \Delta \left({\mathbb{S}}_2^{″}\left(x\right)+R_2\left(x\right)+E_2\left(x\right)\right),$$

(60)

where

$$\begin{array}{cc}\hfill {\mathbb{S}}_2^{″}\left(x\right)=& \sum _{\begin{array}{c}d\le D\\ d\left|P\right(z)\end{array}}{\displaystyle \frac{{\lambda }^{+}\left(d\right)}{\phi \left(d\right)}}\sum _{\begin{array}{c}z<b<x/z\\ (b,d)=1\end{array}}{\displaystyle \frac{\eta \left(b\right)Li(x/2b)}{2}}\hfill \\ \hfill {\mathbb{R}}_2\left(x\right)=& \sum _{\begin{array}{c}d\le D\\ d\left|P\right(z)\end{array}}{\lambda }^{+}\left(d\right)(\sum _{\begin{array}{c}z<b<x/z\\ (b,d)=1\end{array}}\eta \left(b\right)\sum _{\begin{array}{c}{p}_1\sim {\displaystyle \frac{x}{2b}}\\ 2b{p}_2\equiv -1\left(d\right)\\ p_2\equiv 3\left(4\right)\end{array}}1-{\displaystyle \frac{Li(x/2b)}{2\phi \left(d\right)}})\hfill \\ \hfill {\mathbb{E}}_2\left(x\right)=& \sum _{\begin{array}{c}z<b<x/z\\ (b,d)=1\end{array}}\eta \left(b\right)\sum _{\begin{array}{c}d\le D\\ d|{\mathcal{P}}_3\left(z\right)\end{array}}{\lambda }^{+}\left(d\right)\sum _{0<\left|k\right|\le K}c\left(k\right)\sum _{\begin{array}{c}{p}_1\sim {\displaystyle \frac{x}{2b}}\\ 2a{p}_2\equiv -1\left(d\right)\\ p_2\equiv 3\left(4\right)\end{array}}e\left(\alpha b^{\gamma }k{p}_1^{\gamma }\right)\hfill \end{array}$$

We evaluate the sum ${\mathbb{R}}_2\left(x\right)$ in the same way as sum $R_3$ (see (47)) and get

$${\mathbb{R}}_2\left(x\right)\ll {\displaystyle \frac{x}{{(logx)}^A}}.$$

(61)

To estimate sum ${\mathbb{E}}_2\left(x\right)$ we apply Lemma 8 with $r=4$ and get

$${\mathbb{E}}_2\left(x\right)\ll {\displaystyle \frac{x}{{(logx)}^A}}.$$

(62)

With the help of Lemma 1 for sum ${\mathbb{S}}_2^{″}\left(x\right)$ we get

$$\begin{array}{cc}\hfill {\mathbb{S}}_2^{″}\left(x\right)=& \sum _{\begin{array}{c}z<b<x/z\end{array}}{\displaystyle \frac{\eta \left(b\right)Li(x/2b)}{2}}\sum _{\begin{array}{c}d\le D\\ (b,d)=1\end{array}}{\displaystyle \frac{{\lambda }^{+}\left(d\right)}{\phi \left(d\right)}}\hfill \\ \hfill \le & {\displaystyle \frac{x}{4}}\sum _{\begin{array}{c}z<b<x/z\end{array}}{\displaystyle \frac{\eta \left(b\right)}{blog(x/2b)}}\prod _{{\displaystyle \genfrac{}{}{0pt}{}{p|b}{p<z}}}{\left(1-{\displaystyle \frac{1}{p-1}}\right)}^{-1}V\left(z\right)\left(F\left(s\right)+o\left(1\right)\right),\hfill \end{array}$$

where $s={\displaystyle \frac{logD}{logz}}$. Assuming $2<\delta <3$, from $b=n{p}_1$ with $p_1>z$, $n<\frac{x}{z^2}$ and $\frac{x}{z^2}<z$, the conditions $p|b$ and $p<z$ in the above product are equivalent to $p|n$. So, using notation (15), we get

$$\prod _{\begin{array}{c}p|b\\ p<z\end{array}}{\left(1-{\displaystyle \frac{1}{p-1}}\right)}^{-1}=\prod _{p|n}{\left(1-{\displaystyle \frac{1}{p-1}}\right)}^{-1}=h\left(n\right).$$

From $b=n{p}_1$ and $p_1>z$ follows

$$h\left(b\right)=h\left(n\right)\left(1-{\displaystyle \frac{1}{p_1-1}}\right)=h\left(b\right)\left(1+O\left({\displaystyle \frac{1}{z}}\right)\right).$$

So

$${\mathbb{S}}_2^{″}\left(x\right)\le {\displaystyle \frac{x}{4}}V\left(z\right)\left(F\left(s\right)+o\left(1\right)\right)\sum _{\begin{array}{c}z<b<x/z\end{array}}{\displaystyle \frac{\eta \left(b\right)h\left(b\right)}{blog(x/2b)}}.$$

Using Lemma 6, we obtain

$$\begin{array}{cc}\hfill {\mathbb{S}}_2^{″}\left(x\right)\le & {\displaystyle \frac{x}{4}}V\left(z\right)\left(F\left(s\right)+o\left(1\right)\right){\displaystyle \frac{1+o\left(1\right)}{{(logx)}^{1/2}}}·{\displaystyle \frac{A}{2C_1}}\underset{2}{\overset{\delta }{\int }}{\displaystyle \frac{log(t-1)}{t{(1-t/\delta )}^{1/2}}},\hfill \end{array}$$

From (14), (15), (33), it follows that $C_2=C_1{C}_3$ and

$$V\left(z\right)={\displaystyle \frac{2e^{-\gamma }{C}_1{C}_3}{logz}}.$$

Now, with the help of Lemma 3 and choice $D=x^{1/2-\epsilon }$, we receive

$$\begin{array}{cc}\hfill {\mathbb{S}}_2^{″}\left(x\right)\le & {\displaystyle \frac{2e^{-\gamma }{C}_1{C}_{3}x}{4logz}}·{\displaystyle \frac{2e^{\gamma }logz}{logD}}·{\displaystyle \frac{1+o\left(1\right)}{{(logx)}^{1/2}}}·{\displaystyle \frac{A}{2C_1}}{\int }_2^{\delta }{\displaystyle \frac{log(t-1)}{t{(1-t/\delta )}^{1/2}}}\hfill \\ \hfill \le & {\displaystyle \frac{A{C}_{3}x}{2{(logx)}^{1/2}logD}}\underset{2}{\overset{\delta }{\int }}{\displaystyle \frac{log(t-1)}{t{(1-t/\delta )}^{1/2}}}dt\hfill \\ \hfill \le & {\displaystyle \frac{A{C}_{3}x}{(1-2\epsilon ){(logx)}^{3/2}}}\underset{2}{\overset{\delta }{\int }}{\displaystyle \frac{log(t-1)}{t{(1-t/\delta )}^{1/2}}}dt.\hfill \end{array}$$

(63)

From (59)–(63), it follows that

$${\mathbb{S}}_2\left(x\right)\le {\displaystyle \frac{A{C}_3\Delta x}{(1-2\epsilon ){(logx)}^{3/2}}}\underset{2}{\overset{\delta }{\int }}{\displaystyle \frac{log(t-1)}{t{(1-t/\delta )}^{1/2}}}dt.$$

(64)

From (58) and (64), we get

$$\mathbb{S}\left(x\right)\ge {\displaystyle \frac{A{C}_3\Delta x}{{(logx)}^{3/2}}}\left({\displaystyle \frac{1}{\sqrt{2}}}\underset{1}{\overset{\delta /2}{\int }}{\displaystyle \frac{dt}{\sqrt{t(t-1)}}}-{\displaystyle \frac{1}{1-2\epsilon }}\underset{2}{\overset{\delta }{\int }}{\displaystyle \frac{log(t-1)}{t{(1-t/\delta )}^{1/2}}}dt\right).$$

Choosing $\delta =2.0005$ and $\epsilon =0.00001$, we obtain the following.

$$\mathbb{S}\left(x\right)\ge {\displaystyle \frac{c\Delta x}{{(logx)}^{3/2}}}$$

with $c>0$, Theorem 3 is proved.

6. Conclusions

The linear and the 1/2 sieves are powerful tools for studying problems related to prime numbers. These methods play a significant role in number theory, particularly in the distribution of primes and the identification of prime patterns. In this paper, we obtain theorems on the distribution of $\alpha p^{\gamma }+\beta$ modulo one, with primes belonging to two different sets. Based on the progress made in this article, we identify the following open problems:

• Generalization of the obtained results for $\gamma \in (0,1)$.
• Extension of the results to the case $\gamma \in (1,2)$ and to the more complex case $\gamma \in \mathbb{R}\setminus \mathbb{N}$. The main difficulty in this case arises from estimating exponential sums of type (16). The primary requirement for these estimates is that they must be independent of D. Furthermore, there are some difficulties in generalizing the Lemma 10.