Multivariate normal distribution for integral points on varieties

Daniel El-Baz, Daniel Loughran, Efthymios Sofos

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Abstract: Given a variety with coefficients in , we study the distribution of the number of primes dividing the coordinates as we vary an integral point. Under suitable assumptions, we show that this has a multivariate normal distribution. We generalise this to more general Weil divisors, where we obtain a geometric interpretation of the covariance matrix. For our results we develop a version of the Erdős–Kac theorem that applies to fairly general integer sequences and does not require a positive exponent of level of distribution.
Original languageEnglish
Pages (from-to)3089-3128
Number of pages40
JournalTransactions of the American Mathematical Society
Issue number5
Early online date24 Feb 2022
Publication statusPublished - 31 Dec 2022

Bibliographical note

Funding Information:
Received by the editors September 6, 2020, and, in revised form, May 27, 2021. 2020 Mathematics Subject Classification. Primary 14G05; Secondary 60F05, 11N36. The first author was supported by the Austrian Science Fund (FWF), projects F-5512 and Y-901. The second author was supported by EPSRC grant EP/R021422/2.

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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