This extensive article discusses problems related to Kubilius’ fundamental lemma in probabilistic number theory. The author’s idea lies in constructing new couplings of probability spaces supporting dependent and independent random variables. Let \(C_p(n):= C_p(n,m)\) be the multiplicity of the prime factor \(p\) of a random number \(m\leq n\) taken with the probability \(1/n\). By \(Z_p\), \(p\leq n\), we denote independent geometric random variables with \(P(Z_p\geq1)=1/p\). The author proves, in particular, that there exists a joint probability space supporting \(C_p(n)\) and \(Z_p\) such that \[ {\mathbf E}\sum_{p\leq n}\big|C_p(n)-Z_p\big|\leq 2+ O\bigg(\frac{(\log\log n)^2}{\log n}\bigg). \] Similarly, for the ordered sequence of prime factors \(P_1(n)\geq P_2(n)\geq\dots\) of a random \(m\leq n\) and the Poisson-Dirichlet process \((V_1,V_2,\dots)\) with parameter 1, we have \[ {\mathbf E}\sum_i\big|\log P_i(n)-V_i\log n\big|=O(\log\log n). \] The author’s \(\$100\) conjecture asserts that the \(O(1)\) estimate is possible in the last relation.
For the entire collection see [Zbl 0988.00023].