Abstract
Let denote an almost-prime with at most prime factors, counted according to multiplicity. In this paper, it is proved that, for , and , there exist infinitely many primes , such that and , which constitutes an improvement upon the previous result.
1. Introduction and Main Result
Let denote an almost-prime with at most prime factors, counted according to multiplicity. The famous prime twins conjecture states that there exist infinitely many primes such that is a prime too. Up to now, this conjecture is still open, but many approximations about this conjecture were established. One of the most interesting results is due to Chen [1], who showed, in 1973, that there exist infinitely many primes such that .
In 1981, Heath-Brown [2] showed that there exist infinitely many arithmetic progressions of four different terms, three of which are primes and the fourth is . In 2006, Green and Tao [3] established that there exist infinitely many arithmetic progressions consisting of three different primes such that for each . Later, in 2008, Green and Tao [4] showed that, for any , there exist infinitely many arithmetic progressions consisting of different primes such that for each .
Suppose that there is a problem including primes and let be an integer. Having in mind Chen’s result, one may consider the problem with primes , such that . Many authors investigated several kinds of problems of this type, such as Peneva and Tolev [5], Peneva [6], and Tolev [7–9].
Let be an irrational real number and denote the distance from to the nearest integer. Earlier work about the distribution of the fractional parts of the sequence was first considered by Vinogradov [10], who showed that, for any real number , there are infinitely many primes such that for ; then,where denotes arbitrarily small positive number. After that, the first improvement on (1) was due to Vaughan [11], who obtained in (1) and who also required an additional factor on the right-hand side of (1). Since then, many authors improved the upper bound of the exponent , such as Harman [12, 13], Jia [14, 15], and Heath-Brown and Jia [16]. So far, the best result is given by Matomäki [17] with . Moreover, it seems very natural to consider the sequence for , where denotes a prime variable. Also, many authors studied the fractional parts of the sequence for , such as Baker and Harman [18], Harman [19], and Wong [20].
In 2010, Todorova and Tolev [21] considered the distribution of modulo one with primes of the form specified above and showed that, for , there are infinitely many solutions in primes to (1) such that . Later, Matomäki [22] showed that this result actually holds with and . After that, Shi [23] continued to improve the result of Matomäki [22] and showed that there are infinitely many solutions in primes to (1) such that and .
Moreover, for the case , Shi and Wu [24] established the result that there exist infinitely many primes , which satisfy , such that and .
In this paper, we shall continue to improve the result of Shi and Wu [24] and establish the following theorem.
Theorem 1. Suppose that , and . Then, there exist infinitely many primes , which satisfy , such that
Remark 1. According to the work of Shi and Wu [24], our improvement comes from using the methods developed by Tolev [9] with more delicate iterative techniques and various bounds for exponential sums, combining with a version of Lemma 2.2 of [25], while the previous method, in dealing exponential sum, e.g., [24], is based on the traditional pattern of exponential sum estimates.
2. Notation
Let be a sufficiently large real number. Set
Also, we put
Throughout this paper, we always denote primes by and . always denotes an arbitrary small positive constant, which may not be the same at different occurrences. As usual, we use to denote the number of prime factors of counted according to multiplicity, Euler’s function, Möbius’ function, and Mangold’s function, respectively. We denote by the number of solutions of the equation in natural variables . Especially, we write . Let and be the greatest common divisor and the least common multiple of , respectively. Also, we use and , respectively, to denote the integer part of and the distance from to the nearest integer. means that ; means that ; ; . always denotes an almost-prime with at most prime factors, counted according to multiplicity.
3. Preliminary Lemmas
Lemma 1. Let and be any complex numbers. Then, we havewherewhich satisfies
Proof. See Lemma 2.2 of [25].
Lemma 2. Let , and suppose that and that . Assume further that is a complex-valued function. Then, the sumcan be decomposed into sums, each of which is either of Type I:with , , or of Type II:with , .
Lemma 3. For , we have
Proof. See Lemma 4 of Chapter VI of [27].
Lemma 4. Suppose that are real numbers with and that with . Then, we have
Proof. See Lemma 2.2 of [28].
4. Proof of Theorem 1
As shown in [21], we take a periodic function with period 1 such thatwhich has a Fourier series,with coefficients satisfying
The existence of such a function is a consequence of a well-known lemma of Vinogradov. For instance, one can see Chapter I, §2 in [27]. Consider the sumwhere
Let denote the sum of the terms of in which . Then, we have
If we denote by the sum of the terms of in which , it is easy to see that
By noting the fact that the contribution of the terms (if such terms exist) in , for which , is , we deduce thatwhere
On the one hand, if we assume thatthen from (21), we obtainand thus . Hence, there exists a prime , which satisfiesand such that
Combining (13), (25), and (26), we can see that this prime satisfies
On the other hand, by the properties of the weights (for example, one can see Chapter 9 of [29]), it is easy to see that if satisfies (25), thenwhich implies . Therefore, in order to prove Theorem 1, it is sufficient to show that there exists a sequence , which satisfies
By (16) and (18), we can write as follows:where
Next, we shall give lower bound estimate of and upper bound estimate of by using lower bound linear sieve and upper bound linear sieve, respectively. First, we consider . Let be the lower bounds for Rosser’s weights of level . Hence, for any positive integer , there holds
Also, we shall use the fact if , then there holdswhere
Now, we takein (35). By (31) and (34), we obtain
From (32), we have
By (33) and the fact that for , we obtain
Therefore, we obtainwhere
For , by Bombieri–Vinogradov’s mean value theorem (see Chapter 28 of [30]) and (33), we derive that
It follows from Mertens’ prime number theorem (see [31]) that
Then, from (35), (43), and (44), we obtainwhere is defined by (37). For , we shall investigate it in Section 5.
Now, we study the sum , which is defined by (32). We rewrite in the following form:
In order to give upper bound estimate of , we shall apply an upper bound linear sieve. Let be the upper bounds for Rosser’s weights of level . Hence, for any positive integer , we have
Also, we shall use the fact, for , and there holds
For prime in the sum , we take
Then, it is easy to check that , and thus, (49) holds. By (46)–(48), we obtainwhere
If , then . If , by (17) and (52), we know that the representation with and is unique. Thus, it is easy to see that
From (14), we obtain
Thus, we derive thatwhere
By Bombieri–Vinogradov’s mean value theorem and (53), we have
Using (49) and (52), we obtain
Therefore, by (44), (58), and (59), we have
Now, we find a lower bound for the sum . From (30), (38), (41), (45), (51), (56), and (60), we derive thatwhere
Moreover, by partial summation and the prime number theorem, it is easy to show thatwhere
According to simple numerical calculation, we know that
From (44), (61), and (63), we obtain
We shall illustrate that if runs over a suitable sequence, which tends to infinity, then the second error term in (66) can be absorbed. Hence, we need the following lemma.
Lemma 5. Suppose that and are defined in (3) and (4). Let be complex numbers defined for , respectively, which satisfyThen, there exists a sequence satisfying , such that the sum is defined bywhich satisfies
The proof of Lemma 5 will be given in Section 5. From (42) and (57), we know that can be represented as a sum of type (68) withwhere
According to Lemma 5 and (66), there exists a sequence , which tends to infinity, such that
From (44) and (72), we know that there exists a positive constant such that
This completes the proof of Theorem 1.
5. Proof of Lemma 5
In this section, we shall prove Lemma 5. Since , by Dirichlet’s approximation theorem, there exist infinitely many integers and natural numbers with such that
For each such , we choose in a suitable way, i.e., as in (144). In this way, we construct our sequence .
First, we havewhere
According to Lemma 2, by taking , it is easy to see that the sum can be decompose into sums, each of which is either of Type I,with , or of Type IIwith .
Next, we shall deal with the sums of Type I and Type II in the following sections, respectively.
5.1. The Estimate of Type II Sums
In this section, we shall deal with the estimate of the sums of Type II. First, we have
By Cauchy’s inequality, we obtainwhere
If the system of the congruence,has no solution, then . Assume that (82) has a solution. Then, there exists an such that (82) is equivalent to . In this case, we havewhere
The contribution of with to is
Therefore, we have
Moreover, by Cauchy’s inequality again, we obtainwhere
For , we have
Set
Then, we divide into two partswhere denotes the part of which satisfies , while denotes the remaining part of which satisfies . We set in and and derive thatwhere
First, we consider the upper bound for . Let and denote the contribution of the right-hand side of (92) for and , respectively. Trivially, there holds
For , by Lemma 3, we have
By Lemma 4, we have
Combining (92), (93), (96), and (97) and by noting the fact that , we obtain
Now, we consider the estimate of . According to (93), by a splitting argument, we havewhere
By Lemma 1, we havewhere
According to (7) and (101), it is easy to see that
For , we have
It follows from Cauchy’s inequality that
For , we have
By Cauchy’s inequality, we deduce that
For , from Lemma 3, we have
It follows from Lemma 4 that
From (107), (108), and (109), we obtain
Putting (110) into (106), we obtain
Combining (105) and (111), one has
Inserting (112) into (104), we derive thatwhich combines (99) and (103) to obtain
From (91), (98), and (114), we obtainwhich combines (87) yields
5.2. The Estimate of Type I Sums
In this section, we shall deal with the estimate of the sums of Type I. First, we havewhere
By a splitting argument, there holdswhere
For , there exists , which satisfies , such that . Therefore, the equation is equivalent to , i.e., for some . Then, it follows from Cauchy’s inequality that
Set
Then, we havewhere
For , we have
Next, we will discuss the estimate of the right-hand side of (125) in two cases.
Case 1. Suppose that , and under this condition, there holds . By Lemma 4, we haveFrom (119), (123), and (126), we derive that, under the condition , there holds
Case 2. Now, we suppose that . SetApplying Lemma 1 to (125), we havewhereAccording to (7) and (129), it is easy to see thatFor , we haveIt follows from Cauchy’s inequality thatwhereFor , we haveTherefore, by Cauchy’s inequality, one haswhereFor , by Lemma 3, we haveIt follows from Lemma 4 thatFrom (136), (138), and (139), we derive thatwhich combines (133) yieldsFrom (123), (131), and (141), we obtainfrom which and (139), we derive that, under the condition , and there holds
5.3. Proof of Lemma 5
From (116), (127), and (143), by takingthen we deduce that, under conditions (3) and (4), there holdsfor some . This completes the proof of Lemma 5.
Data Availability
The data used to support the findings of the study available within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant nos. 11901566, 12001047, 11971476, and 12071238), Fundamental Research Funds for the Central Universities (Grant no. 2019QS02), and Scientific Research Funds of Beijing Information Science and Technology University (Grant no. 2025035).