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A penalised model reproducing the mod-Poisson fluctuations in the Sathé-Selberg theorem. (English) Zbl 1455.11113

Let \({\mathcal{P}}\) denote the set of primes and let \(\omega(n)\) denote the number of prime divisors of a positive integer \(n\). Roughly speaking, a well-known result of Hardy-Ramanujan from 1916 says that the expected size of \(\omega(n)\) is \(\log\log(n)\). P. Turán in [J. Lond. Math. Soc. 9, 274–276 (1934; Zbl 0010.10401)] and then P. Erdős and M. Kac in [Am. J. Math. 62, 738–742 (1940; Zbl 0024.10203)] refined this result. More precisely, let \(U_n\) denote a uniformly distributed random variable in \(\{1,2,\dots,n\}\). The following statement is the Erdős-Kac Theorem: \[ \sup_{x\in {\mathbb{R}}} |P(\left(\frac{\omega(U_n)-\log\log(n)}{\sqrt{\log\log(n)}}\leq x\right)-\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-u^2/2}\, du|\to 0, \] as \(n\to\infty\). To understand this better, define the \([0,1]\)-Bernoulli random variables \[ B_p^{(n)} = {\mathbf{1}}_{p|U_n}, \] from which it follows that \[ \omega(U_n)=\sum_{p\in {\mathcal{P}}} B_p^{(n)} . \] The Sathé-Selberg theorem, which arose form a series of papers dating from 1953–1954, gives a somewhat complicated asymptotic formula for the expected value of the random variable \(z^{\omega(U_n)}\). The curious reader can find its statement in the paper under review. Define the \(B_p^{(\infty)}\) to be the independent \([0,1]\)-Bernoulli random variables satisfying \[ P(B_p^{(\infty)} = 1) = \frac{1}{p} = 1-P(B_p^{(\infty)} = 0), \] and define \(\Omega_n = \sum_{p\in {\mathcal{P}}, p\leq n} B_p^{(\infty)}\). Note the similarity between \(\Omega_n\) and the previously defined \(\omega(U_n)\). For example, \(\Omega_n\) also satisfies an analog of the Erdős-Kac Theorem.
Now we can, roughly speaking, state what are essentially the motivating questions in the paper under review:
Why doesn’t \(\Omega_n\) have the same approximate probabilistic behaviour as \(\omega(U_n)\) with respect to the Sathé-Selberg theorem?
Is there some way to modify \(\Omega_n\) to obtain consistent probabilistic behaviour?
The author answers both of these interesting questions (the second one affirmatively). The author of the paper under review answers these clearly, precisely and unambiguously. Indeed, his main result takes over a page to even write down. However, in this review, these complicated and technical results are left to the interested reader.

MSC:

11K65 Arithmetic functions in probabilistic number theory
60E10 Characteristic functions; other transforms
60E05 Probability distributions: general theory
60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms
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References:

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