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

TL;DR

The paper investigates consecutive Piatetski-Shapiro primes in two-parameter families, formulating a conjecture for the counting function $\pi(x; (c_1,c_2))$ under the assumption of a strong Hardy-Littlewood framework. It derives a main-term asymptotic $\pi(x; (c_1,c_2)) \sim \dfrac{x^{1/c_1+1/c_2-1}}{c_1 c_2\log x}$ for $c_1\neq c_2$, with a rigorous justification of the main term via a key proposition on weighted averages of singular series, while providing only the main term (no secondary term) due to the complexities of twisted exponential sums. The methodology builds on LO-S’s congruence-modified HL model and extends it to the Piatetski-Shapiro setting, combining intricate exponential-sum analyses with singular-series estimates. The paper also reports numerical data that align with the predicted main term and discusses potential generalizations to longer prime patterns while acknowledging limitations in capturing secondary terms. Overall, the work advances understanding of primes in non-integer power sequences and exemplifies howARCH-type HL conjectures in arithmetic progressions influence the distribution of consecutive primes in exotic integer sequences.

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.

Figures (3)

• Figure 1: $c_1=1.02, c_2 = 1.05$
• Figure 2: $c_1=1.03, c_2 = 1.01$
• Figure 4: The change of the error term for different $c_1$ and $c_2$

Theorems & Definitions (11)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof