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The generalized Bourgain-Sarnak-Ziegler criterion and its application to additively twisted sums on GLm

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Abstract

The Bourgain-Sarnak-Ziegler (BSZ) criterion was initially established for an arbitrary multiplicative function a(n) satisfying |a(n)| ⩽ 1. Recently, Cafferata et al. (2020) showed that the condition |a(n)| ⩽ 1 can be replaced by a(p) 1 to some extent. In this paper, we formulate and prove a further extension of the BSZ criterion, in which the restriction a(p) ⩽ 1 is removed. As applications, we use it together with the analytic theory of automorphic L-functions to prove that there exist some cancellations in the sequence {λπ(n)e(nkα)}n⩽1 on GLm and the Möbius function is disjoint from this sequence.

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Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant No. 11801318) and Natural Science Foundation of Shandong Province (Grant No. ZR2018QA004). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11771252 and 11531008), the Ministry of Education of China (Grant No. IRT16R43), and Taishan Scholars Project. The authors are very grateful to the referees for the very careful reading of the manuscript and helpful suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University, Weihai, 264209, China

    Yujiao Jiang

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

    Guangshi Lü

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  1. Yujiao Jiang
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  2. Guangshi Lü
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Correspondence to Yujiao Jiang.

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Jiang, Y., Lü, G. The generalized Bourgain-Sarnak-Ziegler criterion and its application to additively twisted sums on GLm. Sci. China Math. 64, 2207–2230 (2021). https://doi.org/10.1007/s11425-020-1717-1

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  • DOI: https://doi.org/10.1007/s11425-020-1717-1

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