Carmichael numbers composed of Piatetski-Shapiro primes in Beatty sequences

Jinyun Qi; Victor Zhenyu Guo

doi:10.1515/math-2025-0157

Artikel Open Access

Carmichael numbers composed of Piatetski-Shapiro primes in Beatty sequences

Jinyun Qi und Victor Zhenyu Guo

Veröffentlicht/Copyright: 28. Mai 2025

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Abstract

The Piatetski-Shapiro sequences are sequences of the form ( ⌊ n c ⌋ ) n = 1 ∞ and the Beatty sequence is the sequence of integers ( ⌊ α n + β ⌋ ) n = 1 ∞ . We prove that there are infinitely many Carmichael numbers composed of entirely the primes from the intersection of a Piatetski-Shapiro sequence and a Beatty sequence for c ∈ 1 , 19137 18746 , α > 1 irrational and of finite type by investigating the Piatetski-Shapiro primes in arithmetic progressions in a Beatty sequence. Moreover, we also discuss the intersection of a Piatetski-Shapiro sequence and multiple Beatty sequences in arithmetic progressions.

Keywords: Beatty sequence; Piatetski-Shapiro prime; Carmichael number

MSC 2010: 11N05; 11L07; 11N80; 11B83

1 Introduction

The Piatetski-Shapiro sequences are sequences of the form

N ( c ) ≔ ( ⌊ n c ⌋ ) n = 1 ∞ ( c > 1 , c ∉ N ) ,

where ⌊ t ⌋ denotes the integer part of any t ∈ R . Such sequences have been named in honor of Piatetski-Shapiro [1] who, in 1953, proved that N ( c ) contains infinitely many primes provided that c ∈ ( 1 , 12 11 ) . The range for c in which it is known that N ( c ) contains infinitely many primes has been enlarged many times over the years and is currently known to hold for all c ∈ ( 1 , 243 205 ) , thanks to Rivat and Wu [2].

For fixed real numbers α and β , the associated non-homogeneous Beatty sequence is the sequence of integers defined by

ℬ α , β ≔ ( ⌊ α n + β ⌋ ) n = 1 ∞ ,

which are also called generalized arithmetic progressions. If α is irrational, it follows from a classical exponential sum estimate of Vinogradov [3] that ℬ α , β contains infinitely many prime numbers.

Carmichael numbers are the composite natural numbers N with the property that N ∣ ( a N − a ) for every integer a . In 1994, Alford et al. [4] proved that there exist infinitely many Carmichael numbers. Baker et al. [5] showed that for every c ∈ ( 1 , 147 145 ) , there are infinitely many Carmichael numbers composed entirely of Piatetski-Shapiro primes. Banks and Yeager [6] showed that there are infinitely many Carmichael numbers composed solely of primes from the Beatty sequence ℬ α , β for α , β ∈ R with α > 1 and α is irrational and of finite type.

Since both Piatetski-Shapiro sequences and Beatty sequences produce infinitely many primes, Guo [7] investigated the intersection between a Piatetski-Shapiro sequence and a Beatty sequence by defining

π α , β ( c ) ( x ) ≔ # { p ⩽ x : p ∈ N ( c ) ∩ ℬ α , β }

and derived that

π α , β ( c ) ( x ) = x 1 c α log x + O x 1 c log 2 x ,

for c ∈ ( 1 , 14 13 ) . Later, Guo et al. [8] extend the range of c in this theorem to ( 1 , 12 11 ) .

Guo and Qi [9] considered the following generalized Piatetski-Shapiro sequences:

N α , β ( c ) ≔ ( ⌊ α n c + β ⌋ ) n = 1 ∞

and proved that there are infinitely many Carmichael numbers composed solely of primes from the numbers of the set N α , β ( c ) for c ∈ ( 1 , 64 63 ) .

In this article, we are interested in the relation between Carmichael numbers and the Piatetski-Shapiro primes in a Beatty sequence. For ( a , d ) = 1 , let

π α , β ( c ) ( x ; d , a ) ≔ # { p ⩽ x : p ∈ N ( c ) ∩ ℬ α , β and p ≡ a mod d } .

We prove the following theorem:

Theorem 1.1

Let α ⩾ 1 and β be real numbers. Let α be irrational and of finite type. Let c ∈ 1 , 12 11 and γ ≔ c − 1 .

π α , β ( c ) ( x ; d , a ) = α − 1 γ x γ − 1 π ( x ; d , a ) + α − 1 γ ( 1 − γ ) ∫ 2 x u γ − 2 π ( u ; d , a ) d u + O x 7 13 γ + 11 26 + ε ,

where π ( x ; d , a ) ≔ # { p ⩽ x : p ≡ a mod d } .

Theorem 1.2

Let c ∈ ( 1 , 19137 18746 ) , α be irrational and of finite type. There are infinitely many Carmichael numbers composed of entirely the primes from the set N ( c ) ∩ ℬ α , β .

2 Preliminaries

2.1 Notation

We denote by ⌊ t ⌋ and { t } the integral part and the fractional part of t , respectively. As is customary, we put

e ( t ) ≔ e 2 π i t and { t } ≔ t − ⌊ t ⌋ .

Throughout the article, we make considerable use of the sawtooth function defined by

ψ ( t ) ≔ t − ⌊ t ⌋ − 1 2 = { t } − 1 2 .

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

‖ t ‖ ≔ min n ∈ Z ∣ t − n ∣ .

Let P denote the set of primes in N . The letter p always denotes a prime. For a Beatty sequence ( ⌊ α n + β ⌋ ) n = 1 ∞ , we denote ω ≔ α − 1 . We represent γ ≔ c − 1 for the Piatetski-Shapiro sequence ( ⌊ n c ⌋ ) n = 1 ∞ . We use notation of the form m ∼ M as an abbreviation for M < m ⩽ 2 M .

Throughout the article, ε always denotes an arbitrarily small positive constant, which may not be the same at different occurrences; the implied constants in symbols O , ≪ and ≫ may depend (where obvious) on the parameters α , β , 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 .

2.2 Type of an irrational number

For any irrational number α , we define its type τ = τ ( α ) by the following definition:

τ ≔ sup { t ∈ R : liminf n → ∞ n t ‖ α n ‖ = 0 } .

Using Dirichlet’s approximation theorem, one can see that τ ⩾ 1 for every irrational number α . Thanks to the work of Khintchine [10] and Roth [11,12], it is known that τ = 1 for almost all real numbers, in the sense of the Lebesgue measure, and for all irrational algebraic numbers, respectively. Moreover, if α is an irrational number of type τ < ∞ , then so are α − 1 and n α − 1 for all integer n ⩾ 1 [13].

2.3 Technical lemmas

We need the following well-known approximation of Vaaler [14].

Lemma 2.1

For any H ⩾ 1 , there exist numbers a h , b h such that

ψ ( t ) − ∑ 0 < ∣ h ∣ ⩽ H a h e ( t h ) ⩽ ∑ ∣ h ∣ ⩽ H b h e ( t h ) , a h ≪ 1 ∣ h ∣ , b h ≪ 1 H .

Lemma 2.2

For an arithmetic function g and N ′ ∼ N , we have

∑ N < p ⩽ N ′ g ( p ) ≪ 1 log N max N < N 1 ⩽ 2 N ∑ N < n ⩽ N 1 Λ ( n ) g ( n ) + N 1 ⁄ 2 .

Proof

See the argument on page 48 of [15].□

Lemma 2.3

Suppose that

α = a q + θ q 2 ,

with ( a , q ) = 1 , q ⩾ 1 , ∣ θ ∣ ⩽ 1 . Then there holds

∑ m ⩽ N m ≡ a mod d Λ ( m ) e ( m α ) ≪ N q d − 1 2 + N 4 ⁄ 5 + N 1 ⁄ 2 q 1 ⁄ 2 ( log N ) 3 .

Proof

It is a simplified and weakened version of a theorem of Balog and Perelli [16].□

Lemma 2.4

Suppose that a is a fixed irrational number of finite type τ < ∞ and h ⩾ 1 , m are integers. Then we have

∑ m ⩽ M m ≡ a mod d Λ ( m ) e ( a h m ) ≪ h 1 ⁄ 2 M 1 − 1 ⁄ ( 2 τ ) + ε + M 1 − ε .

Proof

For any sufficiently small ε > 0 , we set ϱ = τ + ε . Since a is of type τ , there exists some constant c > 0 such that

(2.1) ‖ a n ‖ > c n − ϱ , n ⩾ 1 .

For given h with 0 < h ⩽ H , let b ⁄ d be the convergent in the continued fraction expansion of a h , which has the largest denominator d not exceeding M 1 − η for a sufficiently small positive number η . Then we derive that

(2.2) a h − b d ⩽ 1 d M 1 − η ⩽ 1 d 2 ,

which combined with (2.1) yields

M − 1 + η ⩾ ∣ a h d − b ∣ ⩾ ‖ a h d ‖ > c ( h d ) − ϱ .

Taking C 0 ≔ c 1 ⁄ ϱ , we obtain

(2.3) d > C 0 h − 1 M 1 ⁄ ϱ − η ⁄ ϱ .

Combining (2.2) and (2.3), applying Lemma 2.3 and the fact that d ⩽ M 1 − η , we deduce that

∑ m ⩽ M m ≡ a mod d Λ ( m ) e ( a h m ) ≪ ( M d − 1 ⁄ 2 + M 4 ⁄ 5 + M 1 ⁄ 2 d 1 ⁄ 2 ) ( log M ) 3 ≪ ( h 1 ⁄ 2 M 1 − 1 ⁄ ( 2 ϱ ) + η ⁄ ( 2 ϱ ) + M 4 ⁄ 5 + M 1 − η ⁄ 2 ) ( log M ) 3 ≪ h 1 ⁄ 2 M 1 − 1 ⁄ ( 2 τ ) + ε + M 1 − ε .

This completes the proof of Lemma 2.4.□

The following lemma gives a characterization of the numbers in the Beatty sequence ℬ α , β .

Lemma 2.5

A natural number m has the form ⌊ α n + β ⌋ if and only if X α , β ( m ) = 1 , where X α , β ( m ) ≔ ⌊ − α − 1 ( m − β ) ⌋ − ⌊ − α − 1 ( m + 1 − β ) ⌋ .

Proof

Note that an integer m has the form m = ⌊ α n + β ⌋ for some integer n if and only if

□ m − β α ⩽ n < m − β + 1 α .

Finally, we use the following lemma, which provides a characterization of the numbers that occur in the Piatetski-Shapiro sequence N ( c ) .

Lemma 2.6

A natural number m has the form ⌊ n c ⌋ if and only if X ( c ) ( m ) = 1 , where X ( c ) ( m ) ≔ ⌊ − m γ ⌋ − ⌊ − ( m + 1 ) γ ⌋ . Moreover,

X ( c ) ( m ) = γ m γ − 1 + ψ ( − ( m + 1 ) γ ) − ψ ( − m γ ) + O ( m γ − 2 ) .

Proof

The proof of Lemma 2.6 is similar to that of Lemma 2.5, so we omit the details herein.□

Lemma 2.7

For 1 < c < 2817 2426 , there holds

(2.4) π ( c ) ( x ) = ∑ p ⩽ x X ( c ) ( p ) = x γ log x + O x γ log 2 x .

Proof

See Theorem 1 of Rivat and Sargos [17].□

Lemma 2.8

Suppose that

L ( H ) = ∑ i = 1 m A i H a i + ∑ j = 1 n B j H − b j ,

where A i , B j , a i , and b j are positive. Assume further that H 1 ⩽ H 2 . Then there exists some H with H 1 ⩽ H ⩽ H 2 and

L ( H ) ≪ ∑ i = 1 m A i H 1 a i + ∑ j = 1 n B j H 2 − b j + ∑ i = 1 m ∑ j = 1 n ( A i b j B j a i ) 1 ⁄ ( a i + b j ) .

The implied constant depends only on m and n.

Proof

See Lemma 3 of Srinivasan [18].□

Lemma 2.9

For real numbers m 1 , m 2 , and N < t ⩽ N 1 , we have

∑ N < n ⩽ N 1 Λ ( n ) e ( h n γ + m 1 n + m 2 ) ≪ N ε ∣ h ∣ 1 6 N γ 6 + 3 4 + ∣ h ∣ − 1 3 N 1 − γ 3 + ∣ h ∣ 1 4 N γ 4 + 5 8 + ∣ h ∣ − 1 4 N 1 − γ 4 + N 22 25 .

Proof

See [8, Lemma 2.14].□

3 Proof of Theorem 1.1

For a Beatty sequence

ℬ α , β ≔ ⌊ α n + β ⌋ ,

recall that ω ≔ α − 1 . By the definition of π α , β ( c ) ( x ) , we have that

π α , β ( c ) ( x ; d , a ) = ∑ p ⩽ x p ≡ a mod d X α , β ( p ) X ( c ) ( p ) = S 1 + S 2 + S 3 ,

where

S 1 ≔ ∑ p ⩽ x p ≡ a mod d ω X ( c ) ( p ) ; S 2 ≔ ∑ p ⩽ x p ≡ a mod d ( γ p γ − 1 + O ( p γ − 2 ) ) ( ψ ( − ω ( p + 1 − β ) ) − ψ ( − ω ( p − β ) ) ) , S 3 ≔ ∑ p ⩽ x p ≡ a mod d ( ψ ( − ( p + 1 ) γ ) − ψ ( − p γ ) ) ( ψ ( − ω ( p + 1 − β ) ) − ψ ( − ω ( p − β ) ) ) .

A partial summation gives

S 1 = ω γ x γ − 1 π ( x ; d , a ) + ω γ ( 1 − γ ) ∫ 2 x u γ − 2 π ( u ; d , a ) d u + O ( x γ − 1 + 1 ) .

By applying Lemma 2.1, we take H 1 ≔ x ε and let H 2 be chosen later. With a sufficiently small positive number ε , we have that

(3.1) ψ ( − ω ( p + 1 − β ) ) − ψ ( − ω ( p − β ) ) = ∑ 0 < ∣ h 1 ∣ ⩽ H 1 a h 1 ( e ( ω h 1 ( p + 1 − β ) ) − e ( ω h 1 ( p − β ) ) ) + O ∑ ∣ h 1 ∣ ⩽ H 1 b h 1 ( e ( ω h 1 ( p + 1 − β ) ) + e ( ω h 1 ( p − β ) ) )

and

(3.2) ψ ( − ( p + 1 ) γ ) − ψ ( − p γ ) = ∑ 0 < ∣ h 2 ∣ ⩽ H 2 a h 2 ( e ( h 2 ( p + 1 ) γ ) − e ( h 2 p γ ) ) + O ∑ ∣ h 2 ∣ ⩽ H 2 b h 2 ( e ( h 2 ( p + 1 ) γ ) + e ( h 2 p γ ) ) .

We mention that for j = 1 , 2 there holds

a h j ≪ ∣ h j ∣ − 1 and b h j ≪ H j − 1 .

3.1 Upper bounds of S 2

Let N ⩽ x and N 1 ⩽ 2 N . We write S 2 = S 21 + O ( S 22 ) , where

S 21 ≔ ∑ p ⩽ N p ≡ a mod d γ p γ − 1 ( ψ ( − ω ( p + 1 − β ) ) − ψ ( − ω ( p − β ) ) )

and

S 22 ≔ ∑ p ⩽ N p ≡ a mod d γ p γ − 2 ( ψ ( − ω ( p + 1 − β ) ) − ψ ( − ω ( p − β ) ) ) .

By (3.1), Lemma 2.2 and a splitting argument, we obtain that S 21 = S 23 + O ( S 24 ) , where

(3.3) S 23 ≔ ∑ N < n ⩽ N 1 n ≡ a mod d ∑ 0 < ∣ h 1 ∣ ⩽ H 1 a h 1 n γ − 1 Λ ( n ) ( e ( ω h 1 ( n + 1 − β ) ) − e ( ω h 1 ( n − β ) ) )

and

(3.4) S 24 ≔ ∑ N < n ⩽ N 1 n ≡ a mod d ∑ 0 < ∣ h 1 ∣ ⩽ H 1 b h 1 n γ − 1 Λ ( n ) ( e ( ω h 1 ( n + 1 − β ) ) − e ( ω h 1 ( n − β ) ) ) .

First, we estimate S 23 . Let

(3.5) θ h 1 ≔ e ( ω h 1 ) − 1 .

It follows from partial summation and the trivial estimate θ h 1 ≪ 1 that

(3.6) S 23 ≪ ∑ 0 < h 1 ⩽ H 1 a h 1 ∑ N < n ⩽ N 1 n ≡ a mod d n γ − 1 Λ ( n ) θ h 1 e ( ω h 1 ( n − β ) ) ≪ N γ − 1 ∑ 0 < h 1 ⩽ H 1 h 1 − 1 max N 1 ⩽ 2 N ∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) e ( ω h 1 n ) .

Hence, we need to bound

(3.7) T ≔ ∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) e ( ω h 1 n ) .

By Lemma 2.4, we obtain

(3.8) T ≪ h 1 1 2 N 1 − 1 2 τ + ε + N 1 − ε ,

for ε being a small positive number.

Now we work on the bound of S 32 . The contribution of S 24 from h 1 = 0 is

(3.9) 2 b 0 ∑ n ⩽ N n ≡ a mod d Λ ( n ) n γ − 1 ≪ b 0 N γ ϕ ( d ) ≪ H 1 − 1 N γ ,

where the function ϕ ( d ) is the Euler function and b 0 ≪ H 1 − 1 . The contribution from h 1 ≠ 0 is

(3.10) ≪ N γ − 1 H 1 − 1 max N 1 ⩽ 2 N ∑ 0 < h 1 ⩽ H 1 ∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) e ( ω h 1 n ) .

The right-hand side of (3.11) can be estimated by the same method of (3.7). Therefore, by inserting (3.8) into (3.6) and (3.11) and combining with (3.9), it follows that

S 21 ≪ S 23 + S 24 ≪ H 1 1 2 N γ − 1 2 τ + ε + N γ + ε + H 1 − 1 N γ ≪ N γ + ε ,

where we use H 1 = N ε . Moreover, the bound of S 22 can be estimated similarly. Hence, we obtain

(3.11) S 2 ≪ S 21 + S 22 ≪ N γ + ε .

3.2 Upper bounds of S 3

We only give the details of the estimation of S 3 , By (3.1) and (3.2), it is easy to see that

(3.12) S 3 = S 31 + O ( S 32 + S 33 + S 34 ) ,

where

S 31 ≔ ∑ p ⩽ x p ≡ a mod d ∑ 0 < ∣ h 2 ∣ ⩽ H 2 a h 2 ( e ( h 2 ( p + 1 ) γ ) − e ( h 2 p γ ) ) × ∑ 0 < ∣ h 1 ∣ ⩽ H 1 a h 1 ( e ( ω h 1 ( p + 1 − β ) ) − e ( ω h 1 ( p − β ) ) ) , S 32 ≔ ∑ p ⩽ x p ≡ a mod d ∑ 0 < ∣ h 2 ∣ ⩽ H 2 a h 2 ( e ( h 2 ( p + 1 ) γ ) − e ( h 2 p γ ) ) × ∑ ∣ h 1 ∣ ⩽ H 1 b h 1 ( e ( ω h 1 ( p + 1 − β ) ) + e ( ω h 1 ( p − β ) ) ) , S 33 ≔ ∑ p ⩽ x p ≡ a mod d ∑ ∣ h 2 ∣ ⩽ H 2 b h 2 ( e ( h 2 ( p + 1 ) γ ) + e ( h 2 p γ ) ) × ∑ 0 < ∣ h 1 ∣ ⩽ H 1 a h 1 ( e ( ω h 1 ( p + 1 − β ) ) − e ( ω h 1 ( p − β ) ) ) , S 34 ≔ ∑ p ⩽ x p ≡ a mod d ∑ ∣ h 2 ∣ ⩽ H 2 b h 2 ( e ( h 2 ( p + 1 ) γ ) + e ( h 2 p γ ) ) × ∑ ∣ h 1 ∣ ⩽ H 1 b h 1 ( e ( ω h 1 ( p + 1 − β ) ) + e ( ω h 1 ( p − β ) ) ) .

3.2.1 Estimation of S 31

By Lemma 2.2 and a splitting argument, we estimate S 31 by considering

(3.13) ∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) ∑ 0 < ∣ h 2 ∣ ⩽ H 2 a h 2 ( e ( h 2 ( n + 1 ) γ ) − e ( h 2 n γ ) ) × ∑ 0 < ∣ h 1 ∣ ⩽ H 1 a h 1 ( e ( ω h 1 ( n + 1 − β ) ) − e ( ω h 1 ( n − β ) ) ) .

Define

(3.14) ϕ h 2 ( t ) ≔ e ( h 2 ( ( t + 1 ) γ − t γ ) ) − 1 .

Then we have

ϕ h 2 ( t ) ≪ ∣ h 2 ∣ t γ − 1 and ∂ ϕ h 2 ( t ) ∂ t ≪ ∣ h 2 ∣ t γ − 2 .

It follows from the aforementioned estimate, (3.5) and partial summation that the formula (3.13) is

(3.15) ≪ ∑ 0 < ∣ h 2 ∣ ⩽ H 2 1 ∣ h 2 ∣ ∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) ϕ h 2 ( n ) e ( h 2 n γ ) ∑ 0 < ∣ h 1 ∣ ⩽ H 1 a h 1 θ h 1 e ( ω h 1 ( n − β ) ) ≪ ∑ 0 < ∣ h 2 ∣ ⩽ H 2 1 ∣ h 2 ∣ ∫ N N 1 ϕ h 2 ( t ) d ∑ N < n ⩽ t n ≡ a mod d Λ ( n ) e ( h 2 n γ ) ∑ 0 < ∣ h 1 ∣ ⩽ H 1 a h 1 θ h 1 e ( ω h 1 ( n − β ) ) ≪ ∑ 0 < ∣ h 2 ∣ ⩽ H 2 1 ∣ h 2 ∣ ∣ ϕ h 2 ( N ) ∣ ∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) e ( h 2 n γ ) ∑ 0 < ∣ h 1 ∣ ⩽ H 1 a h 1 θ h 1 e ( ω h 1 ( n − β ) ) + ∫ N N 1 ∑ 0 < ∣ h 2 ∣ ⩽ H 2 1 ∣ h 2 ∣ ∂ ϕ h 2 ( t ) ∂ t ∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) e ( h 2 n γ ) ∑ 0 < ∣ h 1 ∣ ⩽ H 1 a h 1 θ h 1 e ( ω h 1 ( n − β ) ) d t ≪ N γ − 1 max N 1 ⩽ 2 N ∑ 0 < ∣ h 2 ∣ ⩽ H 2 ∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) e ( h 2 n γ ) ∑ 0 < ∣ h 1 ∣ ⩽ H 1 a h 1 θ h 1 e ( ω h 1 ( n − β ) ) = N γ − 1 max N 1 ⩽ 2 N ∑ 0 < ∣ h 2 ∣ ⩽ H 2 ∑ 0 < ∣ h 1 ∣ ⩽ H 1 a h 1 θ h 1 ∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) e ( h 2 n γ + ω h 1 n − ω h 1 β ) ≪ N γ − 1 ∑ 0 < ∣ h 1 ∣ ⩽ H 1 1 ∣ h 1 ∣ max N 1 ⩽ 2 N ∑ 0 < ∣ h 2 ∣ ⩽ H 2 ∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) e ( h 2 n γ + ω h 1 n − ω h 1 β ) .

Note that

∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) e ( h 2 n γ + ω h 1 n − ω h 1 β ) = 1 d ∑ m = 1 d ∑ N < n ⩽ N 1 Λ ( n ) e h 2 n γ + ω h 1 n − ω h 1 β + ( n − a ) m d .

Hence, we need to bound

T 1 ≔ ∑ N < n ⩽ N 1 Λ ( n ) e h 2 n γ + ω h 1 + m d n + a m d − ω h 1 β

By Lemma 2.9, we have

(3.16) T 1 N − ε ≪ ∣ h 2 ∣ 1 6 N γ 6 + 3 4 + ∣ h 2 ∣ − 1 3 N 1 − γ 3 + ∣ h 2 ∣ 1 4 N γ 4 + 5 8 + ∣ h 2 ∣ − 1 4 N 1 − γ 4 + N 22 25 .

Recalling H 1 = N ε and inserting (3.16) to (3.15), we have

(3.17) S 31 N − ε ≪ H 2 7 6 N 7 γ 6 − 1 4 + H 2 2 3 N 2 γ 3 + H 2 5 4 N 5 γ 4 − 3 8 + H 2 3 4 N 3 γ 4 + H N γ − 3 25 .

3.2.2 Estimations of S 32 and S 33

We only give the proof of S 32 since the bound of S 33 can be obtained similarly. Let N ⩽ x and N 1 ⩽ 2 N . By Lemma 2.2 and a splitting argument, we can see

S 32 = ∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) ∑ 0 < ∣ h 2 ∣ ⩽ H 2 a h 2 ( e ( h 2 ( n + 1 ) γ ) − e ( h 2 n γ ) ) × ∑ 0 < ∣ h 1 ∣ ⩽ H 1 b h 1 ( e ( ω h 1 ( n + 1 − β ) ) − e ( ω h 1 ( n − β ) ) ) .

By (3.14) and Lemma 2.9, the contribution of S 32 from h 1 = 0 is

(3.18) ≪ H 1 − 1 N γ − 1 ∑ 0 < ∣ h 2 ∣ ⩽ H 2 max N 1 ⩽ 2 N ∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) ( e ( θ h 2 n γ ) ) ≪ H 1 − 1 N γ − 1 ∑ 0 < ∣ h 2 ∣ ⩽ H 2 max N 1 ⩽ 2 N 1 d ∑ m = 1 d ∑ N < n ⩽ N 1 Λ ( n ) e h 2 n γ + m ( n − a ) d ≪ N ε H 2 7 6 N 7 γ 6 − 1 4 + H 2 2 3 N 2 γ 3 + H 2 5 4 N 5 γ 4 − 3 8 + H 2 3 4 N 3 γ 4 + H N γ − 3 25 .

The contribution of S 32 from h 1 ≠ 0 is

∑ p ⩽ x p ≡ a mod d ∑ 0 < ∣ h 2 ∣ ⩽ H 2 a h 2 ( e ( h 2 ( p + 1 ) γ ) − e ( h 2 p γ ) ) × ∑ 0 < ∣ h 1 ∣ ⩽ H 1 b h 1 ( e ( ω h 1 ( p + 1 − β ) ) + e ( ω h 1 ( p − β ) ) ) ,

which can be get the upper bound (3.18) by the same method of S 31 . So we have

(3.19) ( S 32 + S 33 ) N − ε ≪ H 2 7 6 N 7 γ 6 − 1 4 + H 2 2 3 N 2 γ 3 + H 2 5 4 N 5 γ 4 − 3 8 + H 2 3 4 N 3 γ 4 + H N γ − 3 25 .

3.2.3 Estimation of S 34 and conclusions

The contribution of S 34 from h 1 = h 2 = 0 is

(3.20) ∑ p ⩽ N p ≡ a mod d H 2 − 1 H 1 − 1 ≪ H 2 − 1 N 1 + ε .

By (3.14) and Lemma 2.9, the contribution of S 34 from h 1 = 0 and h 2 ≠ 0 is

(3.21) 2 b 0 ∑ p ⩽ x p ≡ a mod d ∑ 0 < ∣ h 2 ∣ ⩽ H 2 b h 2 ( e ( h 2 ( p + 1 ) γ ) + e ( h 2 p γ ) ) ≪ H 1 − 1 H 2 − 1 ∑ 0 < ∣ h 2 ∣ ⩽ H 2 max N 1 ⩽ 2 N ∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) ( e ( h 2 n γ ) ) ≪ N ε H 2 1 6 N γ 6 + 3 4 + H 2 − 1 3 N 1 − γ 3 + H 2 1 4 N γ 4 + 5 8 + H 2 − 1 4 N 1 − γ 4 + N 22 25 ,

where H 1 = N ε and b h j ≪ 1 H . Similarly, by (3.5) and Lemma 2.4, the contribution of S 34 from h 1 ≠ 0 and h 2 = 0 is

(3.22) 2 b 0 ∑ p ⩽ x p ≡ a mod d ∑ 0 < ∣ h 1 ∣ ⩽ H 1 b h 1 ( e ( ω h 1 ( p + 1 − β ) ) + e ( ω h 1 ( p − β ) ) ) ≪ H 1 − 1 H 2 − 1 ∑ 0 < ∣ h 1 ∣ ⩽ H 1 max N 1 ⩽ 2 N ∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) ( e ( ω h 1 n ) ) ≪ H 2 − 1 ( H 1 1 2 N 1 − 1 2 τ + ε + N 1 − ε ) ≪ H 2 − 1 N 1 + ε .

The contribution of S 34 from h 1 ≠ 0 and h 2 ≠ 0 is

∑ N < n ⩽ N 1 n ≡ a mod d Λ ( n ) ∑ 0 < ∣ h 2 ∣ ⩽ H 2 b h 2 ( e ( h 2 ( n + 1 ) γ ) + e ( h 2 n γ ) ) × ∑ 0 < ∣ h 1 ∣ ⩽ H 1 b h 1 ( e ( ω h 1 ( n + 1 − β ) ) + e ( ω h 1 ( n − β ) ) ) ,

which can be estimated similarly. Now the estimation

(3.23) S 34 N − ε ≪ H 2 1 6 N γ 6 + 3 4 + H 2 − 1 3 N 1 − γ 3 + H 2 1 4 N γ 4 + 5 8 + H 2 − 1 4 N 1 − γ 4 + N 22 25 + H 2 − 1 N .

follows from (3.20), (3.21), and (3.22). In the end, by combining (3.17), (3.19), (3.23), (3.12), and (3.11), one has

( S 2 + S 3 ) N − ε ≪ H 2 1 6 N γ 6 + 3 4 + H 2 − 1 3 N 1 − γ 3 + H 2 1 4 N γ 4 + 5 8 + H 2 − 1 4 N 1 − γ 4 + N 22 25 + H 2 7 6 N 7 γ 6 − 1 4 + H 2 2 3 N 2 γ 3 + H 2 5 4 N 5 γ 4 − 3 8 + H 2 3 4 N 3 γ 4 + H N γ − 3 25 + H 2 − 1 N .

By using Lemma 2.8, we obtain

( S 2 + S 3 ) N − ε ≪ N 3 γ 4 + N γ − 3 25 + N 5 γ 4 − 3 8 + N 7 γ 6 − 1 4 + N γ 6 + 3 4 + N γ 4 + 5 8 + N 22 25 + N 7 γ 13 + 11 26 + N 5 γ 9 + 7 18 + N 3 γ 7 + 3 7 + N γ 2 + 11 25 + N γ 7 + 11 14 + N γ 5 + 7 10 .

Note that S 1 ≪ x γ , so we need that S 2 + S 3 ≪ x γ − ε . Hence,

γ > max 9 10 , 5 6 , 22 25 , 11 12 , 7 8 , 3 4 = 11 12

and

S 2 + S 3 ≪ x 7 γ 13 + 11 26 + ε .

4 Sketch of proof of Theorem 1.2

We sketch the proof of Theorem 1.2 because the idea of the proof is close to the proof in [9, Section 4]. We only give the changes that are necessary for our Theorem 1.2.

We set

ϑ ( x ; d , a ) ≔ ∑ p ⩽ x p ≡ a mod d log p

and consider a weighted counting function

ϑ α , β ( c ) ( x ; d , a ) ≔ ∑ p ⩽ x p ∈ π α , β ( c ) ( x ) p ≡ a mod d log p = ∑ p ⩽ x p ≡ a mod d X α , β ( p ) X ( c ) ( p ) log p .

By a similar argument as in the proof of Theorem 1.1, we conclude the following.

Theorem 4.1

Let α ⩾ 1 and β be real numbers. Let c ∈ 1 , 12 11 . Then

ϑ α , β ( c ) ( x ; d , a ) = α − 1 γ x γ − 1 ϑ ( x ; d , a ) + α − 1 γ ( 1 − γ ) ∫ 2 x u γ − 2 ϑ ( u ; d , a ) d u + O x 7 γ 13 + 11 26 + ε .

The proof of our Theorem 1.2 is similar to [9, Section 4] by switching the conditions

1 < c < 14 13 into 1 < c < 12 11 , − 13 35 + 2 γ 5 into − 11 26 + 6 γ 13 , N α , β ( c ) into π α , β ( c )

and

θ into ω .

Let π ( x , y ) be the number of those primes for which p − 1 is free of prime factors exceeding y . Let ℰ be the set of numbers E in the range 0 < E < 1 for which

π ( x , x 1 − E ) ⩾ x 1 + o ( 1 ) ( x → ∞ ) ,

where the function implied by o ( 1 ) depends only on E . By a similar argument as in [5, Page 64–66], we conclude the following statement.

Lemma 4.2

Let α ⩾ 1 and β be real numbers. Let c ∈ 1 , 38 37 . Let B and B 1 be positive real numbers such that B 1 < B < − 11 26 + 6 γ 13 . For any E ∈ ℰ , there is a number x 3 depending on c , B , B 1 , E , and ε , such that for any x ⩾ x 1 , there are at least x E B + ( 1 − B + B 1 ) ( γ − 1 ) − ε Carmichael numbers up to x composed solely of primes from π α , β ( c ) .

Taking B and B 1 arbitrarily close to − 11 26 + 6 γ 13 , Lemma 4.2 implies that there are infinitely many Carmichael numbers composed entirely of the primes from π α , β ( c ) with

− 11 26 + 6 γ 13 E + γ − 1 > 0 .

Taking E = 0.7039 from [19], we eventually have γ > 18746 19137 .

5 More Beatty sequences

Guo et al. [8] proved that there are infinitely many primes in the intersection of a Piatetski-Shapiro sequence and multiple Beatty sequences with some restrictions; see [8, Theorem 1.3] for more details. We mention that by the similar techniques in the proof of Theorem 1.1 and the proof of [8, Theorem 1.3], Piatetski-Shapiro primes in arithmetic progressions and the intersection of multiple Beatty sequences can also be detected. Therefore, we state the following theorem without proofs.

Theorem 5.1

Suppose that ξ is a positive integer, and α 1 , … , α ξ , β 1 , … , β ξ ∈ R . Let α 1 , … , α ξ > 1 be irrational and of finite type such that

1 , α 1 − 1 , … , α ξ − 1 are l i n e a r l y i n d e p e n d e n t o v e r Q .

For c ∈ ( 1 , 12 11 ) , the counting function

π α 1 , β 1 ; … ; α ξ , β ξ ( c ) ( x ; d , a ) ≔ # { prime p ⩽ x : p ≡ a mod d , p ∈ ℬ α 1 , β 1 ∩ … ∩ ℬ α ξ , β ξ ∩ N ( c ) }

satisfies

π α 1 , β 1 ; … ; α ξ , β ξ ( c ) ( x ; d , a ) = α 1 − 1 … α ξ − 1 γ x γ − 1 π ( x ; d , a ) + α 1 − 1 … α ξ − 1 γ ( 1 − γ ) ∫ 2 x u γ − 2 π ( u ; d , a ) d u + O x 7 13 γ + 11 26 + ε ,

where the implied constant depends only on α 1 , … , α ξ and c .

Then by the same technique in the proof of Theorem 1.2, we state the following theorem without proofs.

Theorem 5.2

Suppose that ξ is a positive integer, and α 1 , … , α ξ , β 1 , … , β ξ ∈ R . Let α 1 , … , α ξ > 1 be irrational and of finite type such that

1 , α 1 − 1 , … , α ξ − 1 are l i n e a r l y i n d e p e n d e n t o v e r Q .

For c ∈ ( 1 , 19137 18746 ) , there are infinitely many Carmichael numbers composed entirely of the primes from the set

ℬ α 1 , β 1 ∩ … ∩ ℬ α ξ , β ξ ∩ N ( c ) .

Acknowledgments

The authors express their gratitude to the reviewers for their helpful and detailed comments.

Funding information: The first author was supported in part by the Young Talent Fund of Xi’an Association for Science and Technology (No. 959202413080), the National Science Foundation of Shaanxi Province (No. 2025JC-YBQN-096) and the Undergraduate Talent Cultivation Development Project of Xi’an University (JY2025049). The second author was supported by the National Natural Science Foundation of China (No. 11901447), the Natural Science Foundation of Shaanxi Province (No. 2024JC-YBMS-029), and the Shaanxi Fundamental Science Research Project for Mathematics and Physics (No. 22JSY006).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. All authors wrote the manuscript and are considered to have equal contributions.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

References

[1] I. I. Piatetski-Shapiro, On the distribution of prime numbers in the sequence of the form ⌊f(n)⌋, Mat. Sb. 33 (1953), 559–566. Suche in Google Scholar

[2] J. Rivat and J. Wu, Prime numbers of the form [nc], Glasg. Math. J. 43 (2001), no. 2, 237–254, DOI: https://doi.org/10.1017/S0017089501020080.10.1017/S0017089501020080Suche in Google Scholar

[3] I. M. Vinogradov, A new estimate of a certain sum containing primes (Russian), Rec. Math. Moscou, n. Ser. 2(44) (1937), no. 5, 783–792, English translation: New estimations of trigonometrical sums containing primes, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 17 (1937), 165–166. Suche in Google Scholar

[4] W. R. Alford, A. Granville, and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. (2) 139 (1994), no. 3, 703–722, DOI: https://doi.org/10.2307/2118576. 10.2307/2118576Suche in Google Scholar

[5] R. C. Baker, W. D. Banks, J. Brüdern, I. E. Shparlinski, and A. J. Weingartner, Piatetski-Shapiro sequences, Acta Arith. 157 (2013), no. 1, 37–68, DOI: https://doi.org/10.4064/aa157-1-3. 10.4064/aa157-1-3Suche in Google Scholar

[6] W. C. Banks and A. M. Yeager, Carmichael numbers composed of primes from a Beatty sequence, Colloq. Math. 125 (2011), no. 1, 129–137, DOI: https://doi.org/10.4064/cm125-1-9. 10.4064/cm125-1-9Suche in Google Scholar

[7] V. Z. Guo, Piatetski-Shapiro primes in a Beatty sequence, J. Number Theory 156 (2015), 317–330, DOI: https://doi.org/10.1016/j.jnt.2015.05.001. 10.1016/j.jnt.2015.04.010Suche in Google Scholar

[8] V. Z. Guo, J. Li, and M. Zhang, Piatetski-Shapiro primes in the intersection of multiple Beatty sequences, Rocky Mountain J. Math. 52 (2022), no. 4, 1375–1394. DOI: https://doi.org/10.1216/rmj.2022.52.1375.10.1216/rmj.2022.52.1375Suche in Google Scholar

[9] V. Z. Guo and J. Qi, A generalization of Piatetski-Shapiro sequences, Taiwanese J. Math. 26 (2022), no. 1, 33–47, DOI: https://doi.org/10.11650/tjm/210901. 10.11650/tjm/210802Suche in Google Scholar

[10] A. Khintchine, Zur metrischen Theorie der diophantischen Approximationen, Math. Z. 24 (1926), no. 1, 706–714. 10.1007/BF01216806Suche in Google Scholar

[11] K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1–20, DOI: https://doi.org/10.1112/S0025579300000644. 10.1112/S0025579300000644Suche in Google Scholar

[12] K. F. Roth, Corrigendum to Rational approximations to algebraic numbers, Mathematika 2 (1955), 168. 10.1112/S0025579300000826Suche in Google Scholar

[13] J. Qi and V. Z. Guo, The k-fold divisor function over Beatty sequences, Ramanujan J. 66 (2025), no. 3, 54, DOI: https://doi.org/10.1007/s11139-024-00875-w. 10.1007/s11139-025-01029-2Suche in Google Scholar

[14] J. D. Vaaler, Some extremal functions in Fourier analysis, Bull. Amer. Math. Soc. 12 (1985), no. 2, 183–216, DOI: https://doi.org/10.1090/S0273-0979-1985-15349-2. 10.1090/S0273-0979-1985-15349-2Suche in Google Scholar

[15] S. W. Graham and G. Kolesnik, Van der Corputas Method of Exponential Sums, Cambridge University Press, Cambridge, 1991. 10.1017/CBO9780511661976Suche in Google Scholar

[16] A. Balog and A. Perelli, Exponential sums over primes in an arithmetic progression, Proc. Amer. Math. Soc. 93 (1985), no. 4, 578–582, DOI: https://doi.org/10.1090/S0002-9939-1985-0776182-0. 10.1090/S0002-9939-1985-0776182-0Suche in Google Scholar

[17] J. Rivat and S. Sargos, Nombres premiers de la forme ⌊nc⌋, Canad. J. Math. 53 (2001), no. 2, 414–433, DOI: https://doi.org/10.4153/CJM-2001-017-0. 10.4153/CJM-2001-017-0Suche in Google Scholar

[18] B. R. Srinivasan, The lattice point problem of many-dimensional hyperboloids II, Acta Arith. 8 (1963), 173–204. 10.4064/aa-8-2-173-204Suche in Google Scholar

[19] R. C. Baker and G. Harman, Shifted primes without large prime factors, Acta Arith. 83 (1998), no. 4, 331–361, DOI: https://doi.org/10.4064/aa-83-4-331-361. 10.4064/aa-83-4-331-361Suche in Google Scholar

Received: 2024-08-30

Revised: 2025-04-27

Accepted: 2025-04-28

Published Online: 2025-05-28

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2025-0157

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Beatty sequence; Piatetski-Shapiro prime; Carmichael number

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