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Additive problems on $\lfloor p^c \rfloor$

Lingyu Guo, Victor Zhenyu Guo, Li Lu

Abstract

The sequence $$ \mathbb{P}^{(c)}=(\lfloor p^c \rfloor)_{p\in \mathbb{P}}\quad (c>0,c\notin \mathbb{N}), $$ is an important subsequence of the well-known Piatetski-Shapiro sequence, where $\mathbb{P}$ is the set of prime numbers and $\lfloor \cdot \rfloor$ is the floor function. We prove that for all $c \in (0, 13/15)$, any large enough integer $N$ can be represented as $$ N=\lfloor p^c\rfloor+q, $$ where $p$ and $q$ are primes. We also prove the result holds for almost all fixed positive $c \in \mathbb{R}\setminus\mathbb{Z}$. Moreover, we investigate shifted primes in this sequence, obtaining an asymptotic formula for all $c \in (0, 13/15)$ and an almost-all result for fixed positive $c \in \mathbb{R}\setminus\mathbb{Z}$.

Additive problems on $\lfloor p^c \rfloor$

Abstract

The sequence is an important subsequence of the well-known Piatetski-Shapiro sequence, where is the set of prime numbers and is the floor function. We prove that for all , any large enough integer can be represented as where and are primes. We also prove the result holds for almost all fixed positive . Moreover, we investigate shifted primes in this sequence, obtaining an asymptotic formula for all and an almost-all result for fixed positive .

Paper Structure

This paper contains 21 sections, 12 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Preliminaries
  5. Notation
  6. The explicit formula of $\psi(x)$
  7. The bound for $N(\sigma,T)$ and primes in short intervals
  8. A mean value theorem
  9. The zero-free region
  10. Technical lemmas
  11. Proof of theorem \ref{['thm:main_1']}
  12. An initial construction
  13. The estimates of $S_1$ and $S_3$
  14. The estimate of $S_2$ and the final conclusion
  15. Proof of Theorem \ref{['thm:main_2']}
  16. ...and 6 more sections

Key Result

Theorem 1

For all $c \in (0, 13/15)$ and any fixed integer $a\geqslant 0$, as $x\rightarrow \infty$ we have

Theorems & Definitions (17)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 7 more