Improvements on exponential sums related to Piatetski-Shapiro primes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918239443877888 |
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| author | Lu, Li Guo, Lingyu Guo, Victor Z. |
| author_facet | Lu, Li Guo, Lingyu Guo, Victor Z. |
| contents | We prove a new bound to the exponential sum of the form $$ \sum_{h \sim H}δ_h \mathop{\sum_{m\sim M}\sum_{n\sim N}}_{mn\sim x}a_{m}b_{n}\e\big(αmn + h(mn + u)^γ\big), $$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_11464 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improvements on exponential sums related to Piatetski-Shapiro primes Lu, Li Guo, Lingyu Guo, Victor Z. Number Theory We prove a new bound to the exponential sum of the form $$ \sum_{h \sim H}δ_h \mathop{\sum_{m\sim M}\sum_{n\sim N}}_{mn\sim x}a_{m}b_{n}\e\big(αmn + h(mn + u)^γ\big), $$ 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. |
| title | Improvements on exponential sums related to Piatetski-Shapiro primes |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.11464 |