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arXiv · 2202.06286

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

Victor Z. Guo·Yuan Yi
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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.

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Victor Z. Guo, Yuan Yi. 2022-02-13. Consecutive Piatetski-Shapiro primes based on the Hardy-Littlewood conjecture. https://arxiv.org/abs/2202.06286

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