$k$th powers in a generalization of Piatetski-Shapiro sequences

1.   Introduction

The Piatetski-Shapiro sequences are defined by ${\mathcal{N}}^{(c)} = (\lfloor n^c \rfloor)_{n = 1}^\infty$, where $c > 1$ and $c \notin \mathbb{N}$. In 1953, Piatetski-Shapiro ^[8] proved that ${\mathcal{N}}^{(c)}$ contains infinitely many primes if $c \in (1, \frac{12}{11})$. Moreover, he showed that the prime counting function $\pi^{(c)}(x)$, which counts the number of primes in ${\mathcal{N}}^{(c)}$ up to $x$, satisfies the asymptotic relation $\pi^{(c)}(x) \sim \frac{x^{1/c}}{\log x}$ as $x\to \infty$. Since then, the range of $c$ for which it is known that ${\mathcal{N}}^{(c)}$ contains infinitely many primes has been extended several times, and it is now known that the above formula holds for all $c\in(1, \frac{2817}{2426})$ thanks to the work of Rivat and Sargos ^[10].

This study is related to the topic of Beatty sequences. A non-homogeneous Beatty sequence is a sequence of integers obtained from fixed real numbers $\alpha > 0$ and $\beta$, defined as $\mathcal{B}_{\alpha, \beta} = (\lfloor\alpha n+\beta\rfloor)_{n = 1}^{\infty}$, which is sometimes referred to a generalized arithmetic progression. It is well-known that the sequence contains infinitely many prime numbers if $\alpha$ is irrational ^[4]. Moreover, it is possible to establish an asymptotic relation for the distribution of primes in such sequences, which is the subject of extensive research in number theory. We get that

holds, where $\pi(x)$ is the prime counting function.

The author of this study proposes a generalization of Piatetski-Shapiro sequences in the context of Beatty sequences, which similarly consists of infinitely many prime numbers. Specifically, let $\alpha \geq 1$ and $\beta$ be real numbers. We investigate the following generalized Piatetski-Shapiro sequences:

The Piatetski-Shapiro sequences have deep connections to several fundamental concepts in number theory, such as smooth numbers, square-free numbers and so on. Previous research by Liu, Shparlinski, and Zhang ^[5] focused on the distribution of squares in Piatetski-Shapiro sequences, while Qi, Guo, and Xu ^[9] investigated an intriguing equation related to these sequences. In this paper, we aim to extend their work by exploring the distribution of $k$-th powers in a generalization of Piatetski-Shapiro sequences, which can be viewed as an extension of their previous findings. To be precise, we define $Q_{c, k}^{\alpha, \beta}(s; N)$ as the number of solutions to the equation:

We mention the trivial bound

We prove the following theorem.

Theorem 1.1. Let $k > 1$ be an integer. For any exponent pair $(\kappa, \lambda)$, we have

We also study $Q_{c, k}^{\alpha, \beta}(s; N)$ on average over positive $k$-free integers $s \leq S$. Recall that $\gamma = c^{-1}$and define that

We remark that only the case $S \leq \alpha N^c+\beta$ is meaningful, hence we always assume this. Liu, Shparlinski and Zhang ^[5] showed that

We obtain the following result.

Theorem 1.2. For any $c > 1, c \notin \mathbb{N}$, we have

We remark that the topic is relative to harmonic numbers and their relationships which are interesting mathematical concepts, including number theory, calculus, and physics, as well as the study of degenerate versions or special cases.For further details, refer to ^[6,7].

2.   Preliminaries

2.1.   Notation

We denote by $\left\lfloor{t}\right\rfloor$ and $\{t\}$ the greatest integer ${\leqslant} t$ and the fractional part of $t$, respectively. We also write ${{\bf e}}(t) = e^{2\pi it}$ for all $t\in {\mathbb{R}}$, as usual. We make considerable use of the sawtooth function defined by

In this study, we consider the Piatetski-Shapiro sequence $(\left\lfloor{n^c}\right\rfloor)_{n = 1}^\infty$, where $\left\lfloor{\cdot}\right\rfloor$ denotes the floor function, and $\gamma$ is defined as the inverse of the constant $c$. The set of primes in the natural numbers is denoted by ${\mathbb{P}}$. We use the notation $m\sim M$ to indicate that $m$ lies in the interval $(M, 2M]$.

In order to state our results, we introduce some notation. Throughout the paper, the symbol $\varepsilon$ represents an arbitrarily small positive constant, which may vary from one occurrence to another. The implied constants in the symbols $O$, $\ll$, and $\gg$ may depend on the parameters $c$, $\varepsilon$, $\alpha$, and $\beta$, 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 assertion that the inequality $|F| {\leqslant} \mathcal{C}|G|$ holds with some constant $\mathcal{C} > 0$.

2.2.   Technical lemmas

Lemma 2.1. Let

where $A_i, a_i, B_j, b_j > 0$. Then, (1) for any $Q_2 \geq Q_1 > 0$ there exists $Q \in\left[Q_1, Q_2\right]$ such that

(2) For any $Q_1 > 0$ there exists $Q \in\left(0, Q_1\right]$ such that

Proof. See [3, Lemma 2.4] □

Lemma 2.2. For any $J > 0$, there holds

where

Proof. This is the result of Vaaler ^[11]. □

Lemma 2.3. $\left(\frac{13}{84}+\varepsilon, \frac{55}{84}+\varepsilon\right)$ is an exponent pair.

Proof. See [1, Theorem 6]. □

Lemma 2.4. Let $\alpha, \alpha_1, \alpha_2$ be real constants such that

Let $M, M_1, M_2, x \geq 1$ and let

be two sequences of complex numbers supported on $m \sim M, m_1 \sim M_1$ and $m_2 \sim M_2$ with $\left|\varphi_m\right| \leq 1$. Then, for the sum

we have

Proof. See [2, Theorem 3] □

3.   Proof of Theorem 1.1

Denote $\gamma = c^{-1}$ and $\theta = \alpha^{-\gamma}$. A $k$th power equals $\left\lfloor\alpha n^c+\beta\right\rfloor$ if and only if

which is equivalent to

Let $M = s^{-\frac{1}{k}}\left(\alpha N^c+\beta\right)^{\frac{1}{k}}$, by a normal construction,

where

and

Consider $S_2$. From Lemma 2.2 we have

where

and

for any $J \geq 1$. We begin with $S_3$. Remembering $a_j \ll|j|^{-1}$, we have

Summing over $m$, we obtain

Using the exponent pair $(\kappa, \lambda)$ we get

Then,

which yields

We can readily eliminate the contribution of $S_4$. By utilizing the fact that $b_j$ is bounded by $J^{-1}$, we obtain the following result:

By a similar argument, we have

It yields that

Applying Lemma 2.1 to the bounds on terms in (3.3), we have

Now the result follows from (3.3) and (3.1). Applying the exponent pair $\left(\frac{13}{84}+\varepsilon, \frac{55}{84}+\varepsilon\right)$ from Lemma 2.3 by Bourgain ^[1], people can get the asymptotic formula

4.   Proof of Theorem 1.2

This proof is almost identical to the proof given in [5, Theorem 2.3], so we will provide only a brief outline. We start that

Applying a commonly known result, (see [11, p. 181]),

and a partial summation, we obtain

We proceed as in the proof of Theorem 1.1, so set $\theta = \alpha^{-\gamma}$, $T = \left(\alpha N^c+\beta\right)^{\gamma}$, and

where

and

Using (3.2) and (4.1), we compute $K_0$ directly as follows

By Lemma 2.2

where

and

Using a similar approach in [5, p. 250] we have

where

From Lemma 2.4, we have

Noting that $\gamma > \frac{1}{2}$, it can be easily verified that the fourth term can be combined with the third term on the right-hand side. We can obtain

Hence,

where the term $J^{-1} N^{\frac{c}{k}} S^{1-\frac{1}{k}}$ results from the choice $j = 0$ in the summation on the right-hand side in Lemma 2.2. Now Lemma 2.1 gives

We have the final result.

5.   Conclusions

We have proved Theorems 1.1 and 1.2.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author is supported in part by the Fundamental Research Funds for the Central Universities (No. xzy012021030) and the Shaanxi Fundamental Science Research Project for Mathematics and Physics (No. 22JSY006).

Conflict of interest

We declare no conflict of interest.