Almost primes in Piatetski-Shapiro sequences

: The Piatetski-Shapiro sequences are sequences of the form ( (cid:98) n c (cid:99) ) ∞ n = 1 for c > 1 and c (cid:60) N . It is conjectured that there are inﬁnitely many primes in Piatetski-Shapiro sequences for c ∈ (1 , 2). For every R (cid:62) 1, we say that a natural number is an R -almost prime if it has at most R prime factors, counted with multiplicity. In this paper, we prove that there are inﬁnitely many R -almost primes in Piatetski-Shapiro sequences if c ∈ (1 , c R ) and c R is an explicit constant depending on R .

Such sequences have been named in honor of Piatetski-Shapiro [13], who published the first paper in this problem. He showed that the counting function π (c) (x) . . = prime p x : p ∈ N (c) satisfies the asymptotic relation The range for c of the asymptotic formula of π (c) (x) has been improved by several mathematicians over the years. Kolesnik [7] improved this result to 1 < c < 10 9 = 1.1111 . . . .
Graham did not publish his paper. Heath-Brown [4] applied the Weyl's shift and the exponent pair method, together with his decomposition of the Von Mangoldt function, extended the range to We mention that Leitmann and Wolke [11] proved that the asymptotic formula holds for almost all c ∈ (1, 2) in the sense of Lebesgue measure.
Rivat also considered to prove that there are infinitely many Piatetski-Shapiro primes by giving a lower bound of π (c) (x). He used a sieve method and showed that Later, Baker, Harman and Rivat [1] and Jia [6] improved this range to 1 < c < 20 17 = 1.1764 . . . .
Kumchev [9] improved the range to Eventually, Rivat and Wu [16] gave the best range up to now, which is We remark that if c ∈ (0, 1) then N (c) contains all natural numbers, hence contains all primes. The estimation of Piatetski-Shapiro primes is an approximation of the well-known conjecture that there exists infinitely many primes of the form n 2 + 1. It is conjectured that there are infinitely many Piatetski-Shapiro primes for c ∈ (1, 2). However, the best known bound for c is still far from 2 and the range for c has not been improved for almost 20 years. We approach this problem in a different direction. For every R 1, we say that a natural number is an R-almost prime if it has at most R prime factors, counted with multiplicity. The study of almost primes is an intermediate step to the investigation of primes. In this paper, we prove there are infinitely many almost primes in Piatetski-Shapiro sequences.
Recall that the best known range that there are infinitely many Piatetski-Shapiro primes is (1, 1.1853) and c 3 = 1.1997 > 1.1853. Hence our theorem for 3-almost primes provides a bigger range of c than that of prime numbers. Moreover, when R = 6 we have that which is greater than 2.

Notation
We denote by t and {t} the integer part and the fractional part of t, respectively. As is customary, we put e(t) . . = e 2πit . We make considerable use of the sawtooth function defined by The letter p always denotes a prime. For the Piatetski-Shapiro sequence ( n c ) ∞ n=1 , we denote γ . . = c −1 . We use notation of the form m ∼ M as an abbreviation for M < m 2M.
Throughout the paper, implied constants in symbols O, and may depend (where obvious) on the parameters c, ε but are absolute otherwise. For given functions F and G, the notations F G, G F and F = O(G) are all equivalent to the statement that the inequality |F| C|G| holds with some constant C > 0. F G means that F G F.

The weighted sieve
As we have mentioned the following notion plays a crucial role in our arguments. We specify it to the form that is suited to our applications; it is based on a result of Greaves [3] that relates level of distribution to R-almost primality. More precisely, we say that an N-element set of integers A has a level of distribution D if for a given multiplicative function f (d) we have holds with some real number ρ < R − δ R . Then {a ∈ A : a is an R-almost prime} ρ N log N .
We also need the method of exponent pair. A detailed definition of exponent pair can be found in [2, Page 31]. For an exponent pair (k, l), we denote the A-process and B-process of the exponent pair, respectively.

Initial approach
The set we sieve is A . . = {m x c : m = n c for integer n}.
For any d D, where D is a fixed power of x, we estimate We know that rd ∈ A if and only if rd n c < rd + 1 and rd x.
Within O(1) the cardinality of A d is equal to the number of integers n x for which the interval (n c − 1)d −1 , n c d −1 contains a natural number. Hence By Lemma 2.1 we need to show that for any sufficiently small ε > 0 d D for sufficiently large x. Splitting the range of d into dyadic subintervals, it is sufficient to prove that holds uniformly for D 1 D, N x, N 1 ∼ N. Our aim is to establish (3.1) with D as large as possible. We define We consider S 1 . Writing that φ h . . = e(hd −1 ) − 1 1, we obtain that

Estimation of S by the method of exponent pair
Using the exponent pair (k, l), we have Substituting (3.4) to S 1 and (3.1), it becomes that Now we consider S 2 . The contribution of S 2 from h = 0 is Substituting (3.5) to (3.1), we have The contribution of S 2 from h 0 is which can be estimated by the same method of S 1 . By (3.4), we write (3.6) to be Therefore, to make (3.1) to be true, we need that and