Abstract
Let \(\mathcal {P}_r\) denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we establish a theorem of Bombieri–Vinogradov type for the Piatetski–Shapiro primes \(p=[n^{1/\gamma }]\) with \(\frac{85}{86}<\gamma <1\). Moreover, we use this result to prove that, for \(0.9989445<\gamma <1\), there exist infinitely many Piatetski–Shapiro primes such that \(p+2=\mathcal {P}_3\), which improves the previous results of Lu (Acta Math Sin (Engl Ser) 34(2):255–264, 2018), Wang and Cai (Int J Number Theory 7(5):1359–1378, 2011), and Peneva (Monatsh Math 140(2):119–133, 2003).
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Acknowledgements
The authors would like to express the most sincere gratitude to Professor Wenguang Zhai for his valuable advice and constant encouragement. Also, the authors appreciate the referee for his/her patience in refereeing this paper.
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This work is supported by the National Natural Science Foundation of China (Grant No. 11901566, 12001047, 11971476, 12071238), the Fundamental Research Funds for the Central Universities (Grant No. 2019QS02), and the Scientific Research Funds of Beijing Information Science and Technology University (Grant No. 2025035)
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Li, J., Zhang, M. & Xue, F. An additive problem over Piatetski–Shapiro primes and almost-primes. Ramanujan J 57, 1307–1333 (2022). https://doi.org/10.1007/s11139-021-00390-2
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DOI: https://doi.org/10.1007/s11139-021-00390-2