Almost primes in generalized Piatetski-Shapiro sequences

YUAN Shulei, HUANG Jing, YAN Xiaofei

(College of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China)

Abstract：Let α ≥1, c ≥1 and β be real numbers, we consider a generalization of Piatetski-Shapiro sequences in the sense of Beatty sequences of the formWe define a natural number to be an R-almost prime if it has at most R prime factors for every R ≥1. It is known that there are infinite R-almost primes in sequences if c ∈(1,cR). The aim of this paper is to improve the up bound of cR by choosing exponent pairs.

Keywords： the Piatetski-Shapiro sequences, exponent pair, almost primes

1 Introduction

The Piatetski-Shapiro sequences are of the form

Piatetski-Shapiro has named such sequences and published the first paper in this problem in reference [1-2]. Forc∈(1,2), it is assumed that there are infinitely Piatetski-Shapiro primes. Piatetski-Shapiro has provedholds for 1

Researchers discussed this issue in different directions. If a natural number has at mostRprime factors, we say it is anR-almost prime. The investigation of almost primes is a material step to the study of primes. In reference [5], Baker, Banks, Guo and Yeager proved there are infinitely many primes of the formp=?nc」for any fixedwherenis 8-almost prime. More precisely,

In reference [6], Guo proved that

holds for any sufficiently largex.

Letα≥1 andβbe real numbers, the associated non-homogeneous Beatty sequences are of the form

The generalized Piatetski-Shapiro sequences are of the form

In fact,it can be understood as the rounding of a twice-differentiable function referring reference [7-10]. Letf(x) be a real, twice-differentiable function such that

then {?f(n)」:n∈N} is a generalized Piatetski-Shapiro sequence. Obviously, the values of 2-κandcare corresponding. In this paper, we consider the special case off(x)=αxc+β.

In reference [11], Qi and Xu proved that there exist infinitely many almost primes in sequencesifc∈(1,cR). In this paper, we improve the up bound ofcRby choosing exponent pairs and give corresponding proof. Our result is as follows.

Theorem 1.1For any fixedc∈(1,cR), any realα≥1 and any real numberβ,holds for all sufficiently largex, where the implied constant is absolute. In particular,there have

and

Compared to the result of Qi, this result improvesc3from 1.1997··· to 1.3411··· andc4from 1.6104··· to 1.6105···.

2 Preliminary lemmas

The following notion plays a key role in our debates. We specify it as a form suitable for our applications, it is based on reference [12], which associates level of distribution toR-almost primality. More precisely, we say anN-element set of integersAhas a level of distributionDif for a given multiplicative functionf(d) we obtain

As in reference [12], we have the definition

and

Lemma 2.1LetAbe anN-element set of positive integers, which have a level of distributionDand degreeρ. For some real numberρ

Then

ProofThis is Proposition 1 of Chapter 5 in reference [12].

Lemma 2.2SupposeM≥1 andλare positive real numbers andHbe a positive integer. Iff:[1,M]→R is a real function, which has three continuous derivatives and satisfies

so for the sum

we have

in which the integerMhsatisfies 1 ≤Mh≤Mfor eachh∈[H+1,2H].

ProofThis is Theorem 1 of reference [13].

Lemma 2.3For anyH≥1 we have

with

ProofSee reference [14].

We also need the exponent pair theory, which can be found in reference [15] in detail. For an exponent pair (k,l), we define the A-process and B-process:

and

3 Proof of Theorem 1.1

The proof method mainly comes from Qi and Xu, who adapted the techniques from the original paper of Guo to generalized Piatetski-Shapiro sequences. We shall make some improvements in Sections 3.1-3.3.

Define the set

AsDis a fixed power ofxand for anyd≥D, we estimate

It is clear thatrd∈Aif and only if

The cardinality ofAdis the number of integersn≤xcontaining a natural number in the interval((αnc+β-1)d-1,(αnc+β)d-1] with error termO(1). So

where

By Lemma 2.1 we should prove that

for any sufficiently smallε>0 and sufficiently largex. We divide the range ofd, it is sufficient to verify that

holds uniformly forD1≤D,N≤x,N1N. Then we need to establish (1) withDas large as possible. Let

TakingH=Dxε, by Lemma 2.3 we have

where

and

We splitS1into two parts

where

and

By the exponent pair (k,l), we obtain that

We obtain that

Now we estimateS2. Whenh=0, the contribution ofS2is

Similarly, the contribution ofS2fromh≠0 is

Combining (4)-(7), we get

Therefore, from (1) we need

and

Together with (8) and (9), we get

3.1 Estimation for R=3

By Lemma 2.1,Acontains?x/logxR-almost primes. We use the weighted sieve with the selection

and take

By (10) we need

so

Taking the exponent pair

we obtain

3.2 Estimation for R=4

Similarly, we use the weighted sieve with the selection

and take

By taking the exponent pair

we get

3.3 Estimation for R=5

In this case, we use Lemma 2.2 to estimate (2). From (4) we get

where

Letf(n)=Td-1(n+N)cand

By Lemma 2.2, we have

Hence

To ensure (1) is right, we need that

We use the weighted sieve with the selection

and choose

By Lemma 2.1 and (12) we require

hence

These complete the proof of Theorem 1.1.