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Multivariate normal distribution for integral points on varieties. (English) Zbl 1519.11050

Denote \(\omega(m)\) the number of distinct prime divisors of \(m\). P. Erdős and M. Kac showed in their seminal paper [Am. J. Math. 62, 738–742 (1940; Zbl 0024.10203; JFM 66.0172.02)] that, for any \(\tau\in\mathbb{R}\), the proportion of the integers \(m\le B\) for which \(\omega(m)\le\log\log B+\tau\sqrt{\log\log B}\) tends to the limit \(\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\tau} e^{-t^2/2}dt\) as \(B\to\infty\). In other words, the quantity \(\frac{\omega(m)-\log\log B}{\sqrt{\log\log B}}\) is distributed like a normal distribution with mean \(0\) and variance \(1\).
In the paper under review, for certain given varieties over \(\mathbb{Q}\), the authors study the distribution of the number of prime divisors of the coordinates of varying integral points. The main theorem (Theorem 2.5), which is proved by the method of moments, is a multivariate version of the Erdős-Kac theorem for integer sequences satisfying certain hypotheses. A simplified version (Theorem 2.1) is deduced from the main theorem and the fundamental lemma of the combinatorial sieve [G. Tenenbaum, Introduction to analytic and probabilistic number theory. Transl. from the 2nd French ed. by C. B.Thomas. Cambridge: Cambridge Univ. Press (1995; Zbl 0831.11001)]. Both the statements are too involved to be stated here. For simplicity we cite the most charming special case as follows.
Theorem 1.1. Let \(f\in\mathbb{Z}[x_1,\ldots,x_n]\) be a non-singular homogeneous polynomial with \(n>(\deg f-1)2 ^{\deg f}\). Let \[ \Omega_B=\{\mathbf{x}\in\mathbb{Z}^n: f(\mathbf{x})=0,\, \max_i |x_i|\le B,\,\gcd(x_1,\ldots,x_n)=1\} \] be equipped with the uniform probability measure. If \(f(\mathbf{x})= 0\) has a non-trivial integer solution, then as \(B\to\infty\) the random vectors \[ \Omega_B\to\mathbb{R}^n,\quad \mathbf{x}=(x_1,\ldots,x_n)\mapsto\left(\frac{\omega(x_1)-\log \log B}{\sqrt{\log\log B}},\ldots,\frac{\omega(x_n)-\log\log B}{\sqrt{\log\log B}}\right) \] converge in distribution to the standard multivariate normal distribution on \(\mathbb{R}^n\).
Here the standard multivariate normal distribution means a multivariate normal distribution with zero mean vector and identity covariance matrix.
Reviewer: Ke Gong (Kaifeng)

MSC:

11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
11G35 Varieties over global fields
60F05 Central limit and other weak theorems
11N36 Applications of sieve methods
11N60 Distribution functions associated with additive and positive multiplicative functions
14G05 Rational points
11K65 Arithmetic functions in probabilistic number theory
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