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arXiv · 2504.11464

Improvements on exponential sums related to Piatetski-Shapiro primes

Li Lu·Lingyu Guo·Victor Z. Guo
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Abstract

We prove a new bound to the exponential sum of the form $$ \sum_{h \sim H}\delta_h \mathop{\sum_{m\sim M}\sum_{n\sim N}}_{mn\sim x}a_{m}b_{n}\e\big(\alpha mn + h(mn + u)^{\gamma}\big), $$ by a new approach to the Type I sum. The sum can be applied to many problems related to Piatetski-Shapiro primes, which are primes of the form $\lfloor n^c \rfloor$. In this paper, we improve the admissible range of the Balog-Friedlander condition, which leads to an improvement to the ternary Goldbach problem with Piatetski-Shapiro primes. We also investigate the distribution of Piatetski-Shapiro primes in arithmetic progressions, Piatetski-Shapiro primes in the intersection of multiple Beatty sequences and so on.

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Li Lu, Lingyu Guo, Victor Z. Guo. 2025-04-01. Improvements on exponential sums related to Piatetski-Shapiro primes. https://arxiv.org/abs/2504.11464

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