Consecutive Piatetski-Shapiro primes based on the Hardy-Littlewood conjecture

TL;DR

This paper explores the distribution of consecutive primes within Piatetski-Shapiro sequences, proposing a conjecture based on the Hardy-Littlewood conjecture and providing heuristic and analytical results.

Contribution

It introduces a conjecture on prime pairs in Piatetski-Shapiro sequences supported by heuristic arguments and proves a related proposition on the average of singular series.

Findings

Proposes a conjecture on prime pairs in Piatetski-Shapiro sequences

Provides heuristic support based on Hardy-Littlewood conjecture

Proves a proposition on the average of singular series

Abstract

The Piatetski-Shapiro sequences are of the form ${\mathcal{N}}^{(c)} := (\lfloor n^c \rfloor)_{n=1}^\infty$ with $c > 1, c \not\in \mathbb{N}$. In this paper, we study the distribution of pairs $(p, p^{\#})$ of consecutive primes such that $p \in {\mathcal{N}}^{(c_1)}$ and $p^{\#} \in {\mathcal{N}}^{(c_2)}$ for $c_1, c_2 > 1$ and give a conjecture with the prime counting functions of the pairs $(p, p^{\#})$. We give a heuristic argument to support this prediction which relies on a strong form of the Hardy-Littlewood conjecture. Moreover, we prove a proposition related to the average of singular series with a weight of a complex exponential function.