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Primes in arithmetic progressions with friable indices

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Abstract

We consider the number π(x, y; q, a) of primes p ⩽ x such that p ≡ a (mod q) and (p − a)/q is free of prime factors greater than y. Assuming a suitable form of Elliott-Halberstam conjecture, it is proved that π(x, y; q, a) is asymptotic to ρ(log(x/q)/log y)π(x)/φ(q) on average, subject to certain ranges of y and q, where ρ is the Dickman function. Moreover, unconditional upper bounds are also obtained via sieve methods. As a typical application, we may control more effectively the number of shifted primes with large prime factors.

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Acknowledgements

The first and second authors were supported by the Programme de Recherche Conjoint CNRS-NSFC (Grant No. 1457). The first author was supported by National Natural Science Foundation of China (Grant No. 11531008), the Ministry of Education of China (Grant No. IRT16R43), and the Taishan Scholar Project of Shandong Province. The third author was supported by National Natural Science Foundation of China (Grant No. 11601413) and NSBRP of Shaanxi Province (Grant No. 2017JQ1016).

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Jianya Liu

  2. CNRS LAMA 8050, Université Paris-Est Créteil, Créteil Cedex, 94010, France

    Jie Wu

  3. Department of Mathematics, Xi’an Jiaotong University, Xi’an, 710049, China

    Ping Xi

Authors
  1. Jianya Liu
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  2. Jie Wu
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  3. Ping Xi
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Correspondence to Jie Wu.

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Liu, J., Wu, J. & Xi, P. Primes in arithmetic progressions with friable indices. Sci. China Math. 63, 23–38 (2020). https://doi.org/10.1007/s11425-018-9480-6

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  • DOI: https://doi.org/10.1007/s11425-018-9480-6

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