Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequence

1.   Introduction

Let $q$ be an integer. For each integer $a$ with

we know that ^[1] there exists one and only one $\bar{a}$ with

such that

Define

The work ^[2] posed the problem of investigating a nontrivial estimation for $r(q)$ when $q$ is an odd prime. Zhang ^[3,4] gave several asymptotic formulas for $r(q)$, one of which is:

where $\phi(q)$ is the Euler function and $d(q)$ is the divisor function. Lu and Yi ^[5] studied a generalization of the Lehmer problem over short intervals. Let $n \geqslant 2$ be a fixed positive integer, $q \geqslant 3$ and $c$ be integers with

They defined

where $0 < \theta_1, \theta_2 \leqslant 1$, and obtained

where the $O$ constant depends only on $n$. In addition, Xi and Yi ^[6] considered generalized Lehmer problem over short intervals. Han and Liu ^[7] gave an upper bound estimation for another generalization of the Lehmer problem over incomplete interval.

Guo and Yi ^[8] also found the Lehmer problem has good distribution properties on Beatty sequences. For fixed real numbers $\alpha$ and $\beta$, defined by

Beatty sequences are linear sequences. Based on the results obtained, we conjecture the Lehmer problem also has good distribution properties in some non-linear sequences.

The Piatetski-Shapiro sequence is a non-linear sequence, defined by

where $c \in \mathbb{R}$ is non-integer with $c > 1$ and $z \in \mathbb{R}$. This sequence was first introduced by Piatetski-Shapiro ^[9] to study prime numbers in sequences of the form $\lfloor f(n)\rfloor$, where $f(n)$ is a polynomial. A positive integer is called square-free if it is a product of distinct primes. The distribution of square-free numbers in the Piatetski-Shapiro sequence has been studied extensively. Stux ^[10] found that, as $x$ tends to infinity,

In 1978, Rieger ^[11] improved the range to $1 < c < 3 / 2$ and obtained

Considering the results obtained, we develop this problem by investigating

and range of $c$ when $q$ tends to infinity. By methods of exponential sum and Kloosterman sums and fairly detailed calculations, we get the following result, which is significant for understanding the distribution properties of the Lehmer problem.

Theorem 1.1. Let $q$ be an odd integer and large enough,

we obtain

where the $O$ constant only depends on c.

This paper consists of three main sections. Introduction covers the origins and developments of the Lehmer problem, along with several interesting results. It also presents relevant findings related to the Piatetski-Shapiro sequences. The second section includes some definitions and lemmas throughout the paper. The third section outlines the calculation process, where we use additive characteristics to convert the congruence equations into exponential sum problems. We then employ the Kloosterman sums and exponential sums methods to derive an interesting asymptotic formula.

2.   Preliminary lemmas

To complete the proof of the theorem, we need the following several definitions and lemmas.

In this paper, we denote by $\lfloor t\rfloor$ and $\{t\}$ the integral part and the fractional part of $t$, respectively. As is customary, we put

The notation $\|t\|$ is used to denote the distance from the real number $t$ to the nearest integer; that is,

And $\sum^{\prime}$ indicates that the variable summed over takes values coprime to the number $q$. Throughout the paper, $\varepsilon$ always denotes an arbitrarily small positive constant, which may not be the same at different occurrences; the implied constants in symbols $O, \ll$, and $\gg$ may depend (where obvious) on the parameters $c \text{ and } \varepsilon$, but are absolute otherwise. For given functions $F$ and $G$, the notations

are all equivalent to the statement that the inequality

holds with some constant $\mathcal{C} > 0$.

Lemma 2.1. Let $\mathbf{1}_{c}(m)$ denote the characteristic function of numbers in a Piatetski-Shapiro sequence, then

where

Proof. Note that an integer $m$ has the form

for some integer $n$ if and only if

So

Thie completes the proof. □

Lemma 2.2. Let $H \geqslant 1$ be an integer, $a_h, b_h$ be real numbers, we have

Proof. In 1985, Vaaler showed how Beurling's function could be used to construct a trigonometric polynomial approximation to $\psi (x).$ For each positive integer $N$, Vaaler's construction yields a trigonometric polynomial $\psi^*$ of degree $N$ which satisfies

where

and $\mathrm{sgn\; }(n)$ is the sign of $n$. The Fourier transform $\hat{J}(t)$ satisfies

To be short, we denote

There are more details in Appendix Theorem A.6. of ^[12]. □

Lemma 2.3. Denote

then

where $(m, n, q)$ is the greatest common divisor of $m, n$ and $q$ and $d(q)$ is the number of positive divisors of $q$.

Proof. The proof is given in ^[13]. □

Lemma 2.4. (Korobov ^[14]) Let $\alpha$ be a real number, $Q$ be an integer, and $P$ be a positive integer, then

Lemma 2.5. (Karatsuba ^[15]) For any number $b$, $U < 0$, $K \geqslant 1$, let

then

Lemma 2.6. Suppose $f$ is continuously differentiable, $f^{\prime}(n)$ is monotonic, and

on $I$, then

Proof. See [12, Theorem 2.1]. □

Lemma 2.7. Let $k$ be a positive integer, $k \geqslant 2$. Suppose that $f(n)$ is a real-valued function with $k$ continuous derivatives on $[N, 2N],$ Further suppose that

Then

where the implied constant is absolute.

Proof. See [12, Chapter 3]. □

3.   Proof of theorem

By the definition of Mobius function

it is clear that $n$ is square-free if and only if

where $\omega(n)$ is the number of prime divisor of $n$. So

where

and

3.1.   Estimation of $R_{1}$

From Lemma 2.1, we have

where

Let

and $\lambda(n)$ is Liouville function. When $n \in R(q)$,

We just consider the first term of $R_{11}$. Applying Euler summuation ^[1],

thus

For $R_{12}$, by Lemma 2.2, we have

where

and

Define

then

By Lemma 2.2 and Eq (3.3),

Let

be an exponential pair. Applying Lemma 2.7, it's easy to see

thus

For $R_{122}$, the contribution from $h \neq0$ can be bounded by similar methods of Eq (3.6). Taking

we obtain

It follows from Eqs ($3.5$)–($3.7$),

Hence

3.2.   Estimation of $R_{2}$

Similarly,

where

and

We also just consider the first term of $R_{21}$.

It is easy to see

Since for integers $m$ and $a$, one has

This gives

From Lemma 2.3,

Note the estimate

holds. By Abel summation and Lemma 2.4, we have

To be short, combining Eqs (3.13)–(3.15), we denote

and

Let

making use of Lemma 2.5, we have

Insert it to $R_{211}$, then

By the same method of $R_{211}$,

Following from Eqs (3.10) and (3.11), estimations of $R_{211}$ and $R_{212}$,

By the similar method of $R_{12}$ and $R_{21}$,

where

and

It is obvious that

From the estimate

by partial summation,

For another term of $R_{221}$,

We just need to give an estimation of the last part in (3.20). Similarly, let

then

By partial summation,

where

and

It is obvious that

Suppose $q$ be large enough and for $b > q^{\frac{1}{6}}$, when $2 \nmid d$ or $q \nmid td^2$,

and applying Lemma 2.6, we have

So

which means

We denote

thus,

With Eqs (3.18) and (3.19), we have

For $R_{222}$, the contribution from $h = 0$ can be bounded by similar methods of $R_{21}$, and the contribution from $h \neq 0$ can be bounded by similar methods of $R_{221}$. Taking

we obtain

Following from Eqs (3.16), (3.22), and (3.23),

Hence, from Eqs (3.1), (3.8), and (3.24), we derive that

We need the error terms to be smaller than the main term, so

which means the range of $c$ is $(1, \frac{4}{3})$. The reason why the range of $c$ is changed is that $R(c; q)$ requires $q$ large enough.

4.   Conclusions

In this paper, we generalize the Lehmer problem by considering the count of square-free numbers in the intersection of the Lehmer set and Piatetski-Shapiro sequence when $q$ is an odd integer and large enough. By methods of exponential sum and Kloosterman sum, we study its asymptotic properties and give a sharp asymptotic formula as $q$ tends to infinity.

Based on this result, we will consider some distribution problems similar to the Lehmer problem with more special sequences, which is significant for understanding the distribution properties of those problems.

Author contributions

Xiaoqing Zhao: calculations, writing and editing; Yuan Yi: methodology and reviewing. All authors have read and agreed to the published version of the manuscript.

Use of AI tools declaration

In preparing this manuscript, we employed the language model ChatGPT-4 for the purpose of grammatical corrections. It did not influence the calculations and conclusion in this paper.

Acknowledgments

This work is supported by Natural Science Foundation No.12271422 of China. The authors would like to express their gratitude to the referee for very helpful and detailed comments.

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.