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Explore advanced number theory through this mathematical lecture examining the distribution of prime factors in random integers. Delve into Billingsley's foundational theorem establishing that prime factors of random integers follow the Poisson–Dirichlet distribution of parameter 1, and discover Arratia's innovative proof approach using coupling techniques between prime factors and Poisson–Dirichlet distributions. Learn about recent breakthrough research with Tony Haddad that successfully proves Arratia's long-standing conjecture, providing improved quantitative bounds for this fundamental result. Understand how these sophisticated mathematical methods extend beyond the original problem to yield new insights about the distribution of divisors of integers. Gain exposure to cutting-edge research in analytic number theory, probabilistic methods in mathematics, and the deep connections between random processes and arithmetic functions. This presentation forms part of the Simons Foundation's conference on universal statistics in number theory, offering insight into how probabilistic techniques illuminate the structure of integers and their factorizations.
