@article{11f7a30c8c4548b4bcbb835a16ccc3f2,

title = "Divisibility of the central binomial coefficient (2nn)",

abstract = "We show that for every fixed l ∈ N, the set of n with nl|(2nn) has a positive asymptotic density cl and we give an asymptotic formula for cl as l → ∞. We also show that #{n ≼ x, (n, (2nn)) = 1} ∼ cx/ log x for some constant c. We use results about the anatomy of integers and tools from Fourier analysis. One novelty is a method to capture the effect of large prime factors of integers in general sequences.",

author = "Kevin Ford and Sergei Konyagin",

note = "Funding Information: The first author was supported in part by National Science Foundation Grant DMS-1802139. The authors are grateful to the Institute of Mathematics of the Bulgarian Academy of Sciences, which hosted their visit in June, 2018, and where the seeds of this paper were sown. The first author thanks the Institute of Mathematics at the University of Oxford for providing stimulating working conditions during a visit in March-June, 2019. Funding Information: Received by the editors September 9, 2019, and, in revised form, February 2, 2020. 2020 Mathematics Subject Classification. Primary 05A10, 11B65; Secondary 11N25. The first author was supported in part by National Science Foundation Grant DMS-1802139. The authors thank the anonymous referee for many helpful comments. Publisher Copyright: {\textcopyright} 2020 American Mathematical Society",

year = "2021",

month = feb,

doi = "10.1090/tran/8183",

language = "English (US)",

volume = "374",

pages = "823--953",

journal = "Transactions of the American Mathematical Society",

issn = "0002-9947",

publisher = "American Mathematical Society",

number = "2",

}