Topics in multiplicative and probabilistic number theory

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Multiplication is one of the main arithmetic operations. Understanding its fine properties leads to surprisingly deep questions, and my research program is largely motivated by them. Multiplicative objects often possess a chaotic, random-like behaviour, so that it is natural to study them from a probabilistic point of view.******An important example of a random-like multiplicative object are the primes. These are integers (whole numbers) that cannot be written as product of two other strictly smaller integers. So 2, 3, 5, 7 and 11 are the first few of them. Primes are the "building blocks" of multiplication, since any integer can be written as the product of some of them. Numerical calculations quickly reveal the chaotic nature of primes, with no apparent structure among them. This is one of the main reasons why understanding how primes are distributed among all integers is notoriously hard.******A practical way of packaging many questions about primes is to use multiplicative functions. These are functions that respect the multiplicative structure of the integers. For example, 6 = 2 x 3, so if f is a multiplicative function and we input 6, 2 and 3 to it, the outputs f(6), f(2) and f(3) must obey the same multiplicative law: f(6) = f(2) x f(3). Often, inputing different integers to a multiplicative function can produce outputs that vary unpredictably, thus supplying more examples of "chaotic multiplicative objects". For this reason, we approach them probabilistically and study them on average. Understanding the nature of these averages is one of the main goals of this proposal.******Prime numbers have very simple multiplicative structure, with no non-trivial divisors. The situation is completely different for most other integers. Take, for example, 120: its divisors are the numbers 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. The distribution of the divisors of integers is a key part of this proposal. The divisors of an integer can get very concentrated around certain points, thus forming large clumps. I am particularly interested in understanding how big these clumps can get for a "randomly chosen integer".******The final component of my research proposal concerns rational approximations of irrational numbers. Given a set of denominators, I want to understand whether "most" irrational numbers can be approximated using fractions with these denominators. It is conjectured that the answer is yes if our set of denominators is "large enough". As it turns out, the hardest case is when the denominators have common divisors of very specific size. We thus arrive to another kind of multiplicative structure that needs to be studied.