Almost primes in Piatetski-Shapiro sequences

1.   Introduction

The Piatetski-Shapiro sequences are sequences of the form

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

satisfies the asymptotic relation

if

The range for $c$ of the asymptotic formula of $\pi^{(c)}(x)$ has been improved by several mathematicians over the years. Kolesnik ^[7] improved this result to

Graham and Leitmann ^[10] using the method of exponent pairs independently improved the range to

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

Kolesnik ^[8] using the method of multiple exponential sums improved the range to

Liu and Rivat ^[12] applied the double large sieve to Type I sums and extended the range to

Rivat ^[14] improved the range in his PhD thesis to

Rivat and Sargos ^[15] held the best record to be

We mention that Leitmann and Wolke ^[11] proved that the asymptotic formula holds for almost all $c \in (1, 2)$ 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 $\pi^{(c)}(x)$. He used a sieve method and showed that

if

Later, Baker, Harman and Rivat ^[1] and Jia ^[6] improved this range to

Jia ^[5] extended the range again to

Kumchev ^[9] improved the range to

Eventually, Rivat and Wu ^[16] gave the best range up to now, which is

We remark that if $c\in(0, 1)$ then $\mathcal{N}^{(c)}$ 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 $c \in (1, 2)$. 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 $R \ge 1$, 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.

Theorem 1.1. For any fixed $c \in (1, c_R)$ we have

holds for all sufficiently large $x$. In particular, we have

and

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 $2$.

2.   Preliminaries

2.1.   Notation

We denote by $\left\lfloor {{t}} \right\rfloor$ and $\{t\}$ the integer part and the fractional part of $t$, respectively. As is customary, we put ${\bf{e}}(t): = e^{2\pi it}$. We make considerable use of the sawtooth function defined by

The letter $p$ always denotes a prime. For the Piatetski-Shapiro sequence $(\left\lfloor {{n^c}} \right\rfloor)_{n = 1}^\infty$, we denote $\gamma : = c^{-1}$. We use notation of the form $m\sim M$ as an abbreviation for $M < m\le 2M$.

Throughout the paper, implied constants in symbols $O$, $\ll$ and $\gg$ may depend (where obvious) on the parameters $c, \varepsilon$ but are absolute otherwise. For given functions $F$ and $G$, the notations $F\ll G$, $G\gg F$ and $F = O(G)$ are all equivalent to the statement that the inequality $|F|\le C|G|$ holds with some constant $C > 0$. $F \asymp G$ means that $F \ll G \ll F$.

2.2.   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 $\mathcal{A}$ has a level of distribution $D$ if for a given multiplicative function $f(d)$ we have

As in [3,pp. 174–175] we define the constants

and

We have the following result, which is [3,Chapter 5,Proposition 1].

Lemma 2.1. Suppose $\mathcal{A}$ is an $N$-element set of positive integers with a level of distribution $D$ and degree $\rho$ in the sense that

holds with some real number $\rho < R-\delta_R$. Then

2.3.   Technical lemmas

Lemma 2.2. Let $M \ge 1$ and $\lambda$ be positive real numbers and let $H$ be a positive integer. If $f: [1, M] \to \mathbb{R}$ is a real valued function with three continuous derivatives, which satisfies

then for the sum

where the integer $M_h$ satisfies $1 \le M_h \le M$ for each $h \in [H+1, 2H]$, we have

Proof. See [17,Theorem 1].

Lemma 2.3. For any $H\ge 1$ there are numbers $a_h, b_h$ such that

Proof. See ^[18].

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

and

the A-process and B-process of the exponent pair, respectively.

3.   Proof of Theorem 1.1

3.1.   Initial approach

The set we sieve is

For any $d \le D$, where $D$ is a fixed power of $x$, we estimate

We know that $rd \in \mathcal{A}$ if and only if

Within $O(1)$ the cardinality of $\mathcal{A}_d$ is equal to the number of integers $n \le x$ for which the interval $\big((n^c-1)d^{-1}, n^c d^{-1} \big]$ contains a natural number. Hence

where

By Lemma 2.1 we need to show that for any sufficiently small $\varepsilon > 0$

for sufficiently large $x$. Splitting the range of $d$ into dyadic subintervals, it is sufficient to prove that

holds uniformly for $D_1 \le D, N \le x, N_1 \sim N$. Our aim is to establish (3.1) with $D$ as large as possible. We define

3.2.   Estimation of $S$ by the method of exponent pair

By Lemma 2.3 and taking that

we have

where

and

We consider $S_1$. Writing that

we obtain that

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 $h \neq 0$ 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

Combining (3.7) and (3.8), we obtain that

3.3.   Exponent pair estimation for $R = 3$

We apply the weighted sieve with the choice

and choose

By (3.9) we require that

then by Lemma 2.1, $\mathcal{A}$ contains $\gg x/\log x \; R$-almost primes. To achieve (3.10), we need that

Taking the exponent pair

we have that

3.4.   Exponent pair estimation for $R = 4$

We apply the weighted sieve with the choice

and choose

Similarly to (3.10), we need that

which requires that

by taking the exponent pair

3.5.   A refinement for $R \ge 5$

For $R \ge 5$, we estimate (3.2) by Lemma 2.2. By (3.3) we have that

where

By Lemma 2.2 with $f(n) = Td^{-1} (n+N)^c$ and

it follows that

Hence

The contribution of $S_2$ from $h \neq 0$ can be estimated by the same method and achieve the same upper bound, which means that (3.6) is

Together with the contribution of $S_2$ from $h = 0$, by (3.5) we obtain that the left-hand side of (3.1) is

To ensure the left-hand side of (3.1) is $\ll x^{1- \varepsilon/2}$, we require that

3.6.   The bound of $c$ for $R \ge 5$

We apply the weighted sieve with the choice

and choose

To apply Lemma 2.1, by (3.12) we need that

which gives that

Acknowledgments

The author is supported in part by the National Natural Science Foundation of China (No. 11901447), the China Postdoctoral Science Foundation (No. 2019M653576) and the Natural Science Foundation of Shaanxi Province (No. 2020JQ-009).

Conflict of interest

The author declares no conflicts of interest in this paper.