Diophantine approximation by special primes

TL;DR

This paper proves the existence of infinitely many prime triples with specific linear relations and bounded prime factors in their shifted forms, advancing understanding of Diophantine approximation with primes.

Contribution

It establishes new results on prime triples approximating linear forms with constraints on the number of prime factors of shifted primes.

Findings

Infinitely many prime triples satisfy the inequality with a specific bound.

Each prime in the triple has at most 28 prime factors when increased by 2.

The results depend on certain necessary conditions on the coefficients.

Abstract

We show that whenever $\delta>0$, $\eta$ is real and constants $\lambda_i$ satisfy some necessary conditions, there are infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality $|\lambda_1p_1 + \lambda_2p_2 + \lambda_3p_3+\eta|<(\max p_j)^{-1/12+\delta}$ and such that, for each $i\in\{1,2,3\}$, $p_i+2$ has at most $28$ prime factors.

Equations

$$|{\lambda }_1{p}_1+{\lambda }_2{p}_2+{\lambda }_3{p}_3+\eta |<(\max p_j{)}^{-\xi +\delta }$$

$$n=p_1+p_2+p_3$$

$$p_1+2=P_2^{\prime },p_2+2=P_2^{\prime \prime },p_3+2=P_2^{\prime \prime \prime },$$

$$|{\lambda }_1{p}_1+{\lambda }_2{p}_2+{\lambda }_3{p}_3+\eta |<(\max p_j{)}^{-\xi +\delta }$$

$$p_1+2=P_{28}^{\prime },p_2+2=P_{28}^{\prime \prime },p_3+2=P_{28}^{\prime \prime \prime }.$$

$$\bigg{|}\frac{\lambda_{1}}{\lambda_{2}}-\frac{a_{0}}{q_{0}}\bigg{|}<\frac{1}{q_{0}^{2}}$$

$$X=q_0^{12/5};$$

$$\tau =X^{-5/6}\log X;$$

$$\vartheta =X^{-1/12+\delta },\delta >0;$$

$$H=\frac{{\log }^{2}X}{\vartheta };$$

$$z=X^{\beta },0<\beta <1/30;$$

$$D=X^{47/450-{\epsilon }_0},{\epsilon }_0=0.001;$$

$$\displaystyle P(z)=\prod\limits_{2<p\leq z}p\,,\,\ \mbox{$p$ -prime number}\,;$$

$$I(\alpha )=\underset{{\lambda }_{0}X}{\overset{X}{\int }}e(\alpha y)dy.$$

$$|{\lambda }^{\pm }(d)|\le 1,{\lambda }^{\pm }(d)=0\text{mbox}ifd\ge D\text{mbox}or\mu (d)=0.$$

$$\theta (y)=1\text{mbox}for|y|\le 3\vartheta /4;$$

$$0\le \theta (y)<1\text{mbox}for3\vartheta /4<|y|<\vartheta ;$$

$$\theta (y)=0\text{mbox}for|y|\ge \vartheta .$$

$$\Theta (x)=\underset{-\infty }{\overset{\infty }{\int }}\theta (y)e(-xy)dy$$

$$|\Theta(x)|\leq\min\bigg{(}\frac{7\vartheta}{4},\frac{1}{\pi|x|},\frac{1}{\pi|x|}\bigg{(}\frac{k}{2\pi|x|\vartheta/8}\bigg{)}^{k}\bigg{)}\,.$$

$$\sum _{n\le X}\frac{1}{\phi (n)}\ll \log X.$$

$$\Gamma (X)=\sum _{\genfrac{}{}{0ex}{}{{\lambda }_{0}X<p_1,p_2,p_3\le X}{\genfrac{}{}{0ex}{}{|{\lambda }_1{p}_1+{\lambda }_2{p}_2+{\lambda }_3{p}_3+\eta |<\vartheta }{(p_i+2,P(z))=1,i=1,2,3}}}\log p_1\log p_2\log p_3.$$

$$\Gamma (X)\ge \tilde{\Gamma }(X)=\sum _{\genfrac{}{}{0ex}{}{{\lambda }_{0}X<p_1,p_2,p_3\le X}{(p_i+2,P(z))=1,i=1,2,3}}\theta ({\lambda }_1{p}_1+{\lambda }_2{p}_2+{\lambda }_3{p}_3+\eta )\log p_1\log p_2\log p_3.$$

$$\tilde{\Gamma }(X)=\sum _{{\lambda }_{0}X<p_1,p_2,p_3\le X}\theta ({\lambda }_1{p}_1+{\lambda }_2{p}_2+{\lambda }_3{p}_3+\eta ){\Lambda }_1{\Lambda }_2{\Lambda }_3\log p_1\log p_2\log p_3,$$

$${\Lambda }_i=\sum _{d|(p_i+2,P(z))}\mu (d),i=1,2,3$$

$${\Lambda }_i^{\pm }=\sum _{d|(p_i+2,P(z))}{\lambda }^{\pm }(d),i=1,2,3$$

$${\Lambda }_1{\Lambda }_2{\Lambda }_3\ge {\Lambda }_1^{-}{\Lambda }_2^{+}{\Lambda }_3^{+}+{\Lambda }_1^{+}{\Lambda }_2^{-}{\Lambda }_3^{+}+{\Lambda }_1^{+}{\Lambda }_2^{+}{\Lambda }_3^{-}-2{\Lambda }_1^{+}{\Lambda }_2^{+}{\Lambda }_3^{+}$$

$$\tilde{\Gamma }(X)$$

$$\times ({\Lambda }_1^{-}{\Lambda }_2^{+}{\Lambda }_3^{+}+{\Lambda }_1^{+}{\Lambda }_2^{-}{\Lambda }_3^{+}+{\Lambda }_1^{+}{\Lambda }_2^{+}{\Lambda }_3^{-}-2{\Lambda }_1^{+}{\Lambda }_2^{+}{\Lambda }_3^{+})\log p_1\log p_2\log p_3.$$

$${\Gamma }_0(X)={\Gamma }_1(X)+{\Gamma }_2(X)+{\Gamma }_3(X)-2{\Gamma }_4(X),$$

$${\Gamma }_1(X)=\sum _{{\lambda }_{0}X<p_1,p_2,p_3\le X}\theta ({\lambda }_1{p}_1+{\lambda }_2{p}_2+{\lambda }_3{p}_3+\eta ){\Lambda }_1^{-}{\Lambda }_2^{+}{\Lambda }_3^{+}\log p_1\log p_2\log p_3$$

$${\Gamma }_1(X)=\sum _{\genfrac{}{}{0ex}{}{d_i|P(z)}{i=1,2,3}}{\lambda }^{-}(d_1){\lambda }^{+}(d_2){\lambda }^{+}(d_3)\sum _{\genfrac{}{}{0ex}{}{{\lambda }_{0}X<p_1,p_2,p_3\le X}{p_i+2\equiv 0(d_i),i=1,2,3}}\theta ({\lambda }_1{p}_1+{\lambda }_2{p}_2+{\lambda }_3{p}_3+\eta )\log p_1\log p_2\log p_3.$$

$${\Gamma }_1(X)$$

$$\times \underset{-\infty }{\overset{\infty }{\int }}\Theta (t)e(({\lambda }_1{p}_1+{\lambda }_2{p}_2+{\lambda }_3{p}_3+\eta )t)dt$$

$$=\underset{-\infty }{\overset{\infty }{\int }}\Theta (t)e(\eta t)L^{-}({\lambda }_{1}t,X)L^{+}({\lambda }_{2}t,X)L^{+}({\lambda }_{3}t,X)dt,$$

$$L^{\pm }(t,X)=\sum _{d|P(z)}{\lambda }^{\pm }(d)\sum _{\genfrac{}{}{0ex}{}{{\lambda }_{0}X<p\le X}{p+2\equiv 0(d)}}e(pt)\log p.$$

Full text

Keywords: Rosser’s weights, vector sieve, circle method.

2000 Math. Subject Classification. 11D75, 11N36, 11P32.

1 Introduction and statements of the result.

In 1973 Vaughan [13] proved that whenever $\delta>0$, $\eta$ is real and constants $\lambda_{i}$ satisfy some necessary conditions, there are infinitely many prime triples $p_{1},\,p_{2},\,p_{3}$ such that

[TABLE]

for $\xi=1/10$. Latter the upper bound for $\xi$ was improved by Baker and Harman [1] to $\xi=1/6$, by Harman [4] to $\xi=1/5$ and the best result up to now is due to K. Matomäki [9] with $\xi=2/9$.

On the other hand a famous and still unsolved problem in Number Theory is the prime-twins conjecture, which states that there exist infinitely many prime numbers $p$ such that $p+2$ is also a prime.

Up to now many hybrid theorems were proved. One of the best result belongs to K. Matomäki and Shao [8]. They proved that every sufficiently large odd integer $n$ such that $n\equiv 3\,\pmod{6}$ can be represented as a sum

[TABLE]

of primes $p_{1},\,p_{2},\,p_{3}$ such that

[TABLE]

where $P_{l}$ is a number with at most $l$ prime factors.

In the present paper we consider (1) with primes of the form specified above. We prove the following theorem.

Theorem 1**.**

Suppose that $\lambda_{1},\lambda_{2},\lambda_{3}$ are non-zero real numbers, not all of the same sign, that $\eta$ is real, and that $\lambda_{1}/\lambda_{2}$ is irrational. Let $\xi=1/12$ and $\delta>0$. Then there are infinitely many ordered triples of primes $p_{1},\,p_{2},\,p_{3}$ for which

[TABLE]

and

[TABLE]

By choosing the parameters in a different way we may obtain other similar results, for example $\xi=-9/350\,,\,p_{i}+2=P_{20}\,,i=1,2,3$.

Result of this type were obtained by Dimitrov and Todorova [3]. Combining the circle and sieve methods and using the Bombieri – Vinogradov prime number theorem they proved (2) with right-hand side $[\log(\max p_{j})]^{-A}$, $A>1$ and primes $p_{1},\,p_{2},\,p_{3}$ such that $p_{i}+2=P_{8}\,,i=1,2,3$. In this paper we improve the right-hand side of [3]. Obviously this is at the expense of the number of the prime factors of $p_{i}+2$.

2 Notations and some lemmas.

For positive $A$ and $B$ we write $A\asymp B$ instead of $A\ll B\ll A$. As usual $\varphi(n)$ and $\mu(n)$ denote Euler’s function and Möbius’ function. Let $(m_{1},m_{2})$ and $[m_{1},m_{2}]$ be the greatest common divisor and the least common multiple of $m_{1},m_{2}$ respectively. Instead of $m\equiv n\,\pmod{k}$ we write for simplicity $m\equiv n(k)$. As usual, $[y]$ denotes the integer part of $y$, $e(y)=e^{2\pi\imath y}$. The letter $\varepsilon$ denotes an arbitrary small positive number, not the same in all appearances. For example this convention allows us to write $x^{\varepsilon}\log x\ll x^{\varepsilon}$. Since $\lambda_{1}/\lambda_{2}$ is irrational, there are infinitely many different convergents $a_{0}/q_{0}$ to its continued fraction, with

[TABLE]

where $(a_{0},q_{0})=1,q_{0}\geq 1$ and $a_{0}\neq 0$ . We choose $q_{0}$ to be large in terms of $\lambda_{1},\lambda_{2},\lambda_{3}$ and $\eta$, and make the following definitions.

[TABLE]

The value of $\beta$ will be specified latter.

Let $\lambda^{\pm}(d)$ be the lower and upper bounds Rosser’s weights of level $D$, hence

[TABLE]

For further properties of Rosser’s weights we refer to [5], [6].

Lemma 1**.**

Let $\vartheta\in\mathbb{R}$ and $k\in\mathbb{N}$. There exists a function $\theta(y)$ which is $k$ times continuously differentiable and such that

[TABLE]

and its Fourier transform

[TABLE]

satisfies the inequality

[TABLE]

Proof.

See [10]. ∎

Lemma 2**.**

Let $X\geq 2$, $k\geq 2$. We have

[TABLE]

3 Outline of the proof.

Consider the sum

[TABLE]

Any non-trivial estimate from below of $\Gamma(X)$ implies solvability of $|\lambda_{1}p_{1}+\lambda_{2}p_{2}+\lambda_{3}p_{3}+\eta|<\vartheta$ in primes such that $p_{i}+2=P_{h},\;h=[\beta^{-1}]$.

We have

[TABLE]

On the other hand

[TABLE]

where

[TABLE]

We denote

[TABLE]

From the linear sieve we know that $\Lambda_{i}^{-}\leq\Lambda_{i}\leq\Lambda_{i}^{+}$ (see [2], Lemma 10). Then we have a simple inequality

[TABLE]

analogous to this one in ([2], Lemma 13).

Using (15) and (17) we obtain

[TABLE]

Let

[TABLE]

where for example

[TABLE]

and so on. We shall consider the sum $\Gamma_{1}(X)$. The rest can be considered in the same way.

From (16) and (20) we have

[TABLE]

Using the inverse Fourier transform for the function $\theta(x)$ we get

[TABLE]

where

[TABLE]

We divide $\Gamma_{1}(X)$ into three parts

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

We shall estimate $\Gamma_{1}^{(3)}(X)$, $\Gamma_{1}^{(1)}(X)$, $\Gamma_{1}^{(2)}(X)$ respectively in the sections 4, 5, 6. In section 7 we shall complete the proof of the Theorem.

4 Upper bound for $\mathbf{\Gamma_{1}^{(3)}(X)}$.

Lemma 3**.**

For the integral $\Gamma_{1}^{(3)}(X)$, defined by (25), we have

[TABLE]

Proof.

See [[3], Lemma 2]. ∎

5 Asymptotic formula for $\mathbf{\Gamma_{1}^{(1)}(X)}$.

The first lemma we need in this section is the following.

Lemma 4**.**

Let $\lambda\neq 0$. Using the definitions (11) and (21) we have

[TABLE]

Proof.

We only prove (i). The inequality (ii) can be proved likewise.

Having in mind (5), (12) and (21) we get

[TABLE]

where

[TABLE]

We have

[TABLE]

Obviously $V\leq\sum\limits_{l}V_{l}$ where

[TABLE]

and $l$ takes the values $2^{d}/\tau,\,d=0,1,2,...$, with $l\leq X$.

We have

[TABLE]

The assertion in (i) follows from (5), (5) – (5). ∎

The next lemma gives us asymptotic formula for the sums $L^{\pm}(\alpha,X)$ denoted by (21).

Lemma 5**.**

Let $D$ is defined by (9), and $\lambda(d)$ be complex numbers defined for $d\leq D$ such that

[TABLE]

If

[TABLE]

then for $|\alpha|\leq\tau$ we have

[TABLE]

where $A>0$ is an arbitrary large constant.

Proof.

This lemma is very similar to results of Tolev [12]. Inspecting the arguments presented in ([12], Lemma 10), the reader will easily see that the proof of Lemma 5 can be obtained by the same manner. ∎

Let

[TABLE]

We use the identity

[TABLE]

Replace

[TABLE]

Then from (21), (23), (5), (5), (35), Lemma 1 and Lemma 5 we obtain

[TABLE]

On the other hand (5) and Lemma 2 give us

[TABLE]

Bearing in mind (5), (37) and Lemma 4 we find

[TABLE]

Arguing as in [3] for the integral defined by (35) we get

[TABLE]

where

[TABLE]

According to ([3], Lemma 4) we have

[TABLE]

Usung (5), (6), (38) and (39) we obtain

[TABLE]

Let

[TABLE]

Thus from (41) and (42) it follows

[TABLE]

6 Upper bound for $\mathbf{\Gamma_{1}^{(2)}(X)}$.

The treatment of the intermediate region depends on the following lemma.

Lemma 6**.**

Suppose $\alpha\in\mathbb{R}$ with a rational approximation $\dfrac{a}{q}$ satisfying \displaystyle\bigg{|}\alpha-\frac{a}{q}\bigg{|}<\frac{1}{q^{2}}, where $(a,q)=1,q\geq 1$, $a\neq 0$. Let $D$ is defined by (9), and $\omega(d)$ be complex numbers defined for $d\leq D$ and let $\omega(d)\ll 1$. If

[TABLE]

then

[TABLE]

where $\varepsilon$ is an arbitrary small positive number.

Proof.

See [[11], Lemma 1]. ∎

Let us consider any sum $L^{\pm}(\alpha,\,X)$ denoted by (21). We represent it as sum of finite number sums of the type

[TABLE]

where

[TABLE]

We have

[TABLE]

If

[TABLE]

then from Lemma 6 for the sums $L(\alpha,\,Y)$ we get

[TABLE]

Therefore

[TABLE]

Let

[TABLE]

We shall prove the following

Lemma 7**.**

Let $t,\,X,\,\lambda_{1},\,\lambda_{2}\in\mathbb{R}$,

[TABLE]

where $\tau$ and $H$ are denoted by (5) and (7), $\lambda_{1}/\lambda_{2}\in\mathbb{R}\backslash\mathbb{Q}$ and $V(t,\,X)$ is defined by (47). Then there exists a sequence of real numbers $X_{1},\,X_{2},\ldots\to\infty$ such that

[TABLE]

Proof.

Our aim is to prove that there exists a sequence $X_{1},\,X_{2},\,...\to\infty$ such that for each $j=1,2,\ldots$ at least one of the numbers $\lambda_{1}t$ and $\lambda_{2}t$ with t, subject to (48) can be approximated by rational numbers with denominators, satisfying (45). Then the proof follows from (46) and (47).

Let $q_{0}$ be sufficiently large and $X$ be such that $X=q_{0}^{12/5}$ (see (4)). Let us notice that there exist $a_{1},\,q_{1}\in\mathbb{Z}$, such that

[TABLE]

From Dirichlet’s Theorem ([7], p.158) it follows the existence of integers $a_{1}$ and $q_{1}$, satisfying the first three conditions. If $a_{1}=0$ then $|\lambda_{1}t|<\displaystyle\frac{1}{q_{1}q_{0}^{2}}$ and from (48) it follows

[TABLE]

From the last inequality, (4) and (5) we obtain

[TABLE]

which is impossible for large $q_{0}$, respectively, for a large $X$. So $a_{1}\neq 0$. By analogy there exist $a_{2},\,q_{2}\in\mathbb{Z}$, such that

[TABLE]

If q_{i}\in\bigg{[}X^{1/6},\,X^{5/6}\bigg{]} for $i=1$ or $i=2$, then the proof is completed. From (4), (50) and (51) we have

[TABLE]

Thus it remains to prove that the case

[TABLE]

is impossible. Let $q_{i}<X^{1/6}$, $i=1,2$. From (7), (48), (50) – (52) it follows

[TABLE]

We have

[TABLE]

where \mathfrak{T}_{i}=\dfrac{q_{i}}{a_{i}}\bigg{(}\lambda_{i}t-\dfrac{a_{i}}{q_{i}}\bigg{)}\,,\;i=1,\,2. According to (50), (51) and (54) we obtain

[TABLE]

Thus $\;\displaystyle\frac{a_{1}q_{2}}{a_{2}q_{1}}=\mathcal{O}(1)$ and

[TABLE]

Therefore, both fractions $\displaystyle\frac{a_{0}}{q_{0}}$ and $\displaystyle\frac{a_{1}q_{2}}{a_{2}q_{1}}$ approximate $\displaystyle\frac{\lambda_{1}}{\lambda_{2}}$. Using (50), (52) and inequality (53) with $i=2$ we obtain

[TABLE]

so $|a_{2}|q_{1}\neq q_{0}$ and the fractions $\displaystyle\frac{a_{0}}{q_{0}}$ and $\displaystyle\frac{a_{1}q_{2}}{a_{2}q_{1}}$ are different. Then using (56) it follows

[TABLE]

On the other hand, from (3) and (55) we have

[TABLE]

which contradicts (57). This rejects the assumption (52). Let $q_{0}^{(1)},\,q_{0}^{(2)},\,\ldots$ be an infinite sequence of values of $q_{0}$, satisfying (3). Then using (4) one gets an infinite sequence $X_{1},\,X_{2},\,\ldots$ of values of $X$, such that at least one of the numbers $\lambda_{1}t$ and $\lambda_{2}t$ can be approximated by rational numbers with denominators, satisfying (45). Hence, the proof is completed. ∎

Let us estimate the integral $\Gamma_{1}^{(2)}(X_{j})$, denoted by (24). Using (47), (49) and Lemma 1 we find

[TABLE]

where

[TABLE]

Arguing as in [3] we obtain

[TABLE]

Using (6), (59) and choosing $\varepsilon<\delta$ we get

[TABLE]

Summarizing (22), (26), (43) and (60) we find

[TABLE]

7 Proof of the Theorem.

Since the sums $\Gamma_{2}(X_{j})$, $\Gamma_{3}(X_{j})$ and $\Gamma_{4}(X_{j})$ are estimated in the same way then from (13), (14), (3), (19) and (61) we obtain

[TABLE]

where

[TABLE]

and $G^{\pm}$ are defined by (42).

Let $f(s)$ and $F(s)$ are the lower and the upper functions of the linear sieve. We know that if

[TABLE]

then

[TABLE]

where $\gamma=0.577...$ is the Euler constant (see Lemma 10,[2]). Using (42) and Lemma 10 [1] we get

[TABLE]

Here

[TABLE]

To estimate $W(X_{j})$ from below we shall use the inequalities (see (7)):

[TABLE]

Let $X=X_{j}$. Then from (63) and (68) it follows

[TABLE]

Choose $s=2.948$.

Then by (8), (9) and (64) we find

[TABLE]

It is not difficult to compute that for sufficiently large $X$ we have

[TABLE]

Choose $A\geq 10$.

Then by (6), (40), (62), (67), (69) and (70) we obtain:

[TABLE]

The last inequality implies that $\Gamma(X_{j})\rightarrow\infty$ as $X_{j}\rightarrow\infty$.

By the definition (13) of $\Gamma(X)$ and the inequality (71) we conclude that for some constant $c_{0}>0$ there are at least $c_{0}\frac{X_{j}^{23/12+\delta}}{(\log X_{j})^{6}}$ triples of primes $p_{1},p_{2},p_{3}$ satisfying $\lambda_{0}X_{j}<p_{1},p_{2},p_{3}\leq X_{j},\;|\lambda_{1}p_{1}+\lambda_{2}p_{2}+\lambda_{3}p_{3}+\eta|<\vartheta$ and such that for every prime factor $p$ of $p_{j}+2,\,j=1,2,3$ we have $p\geq X^{0.035089}$.

The proof of the Theorem is complete.