Topics in Multiplicative and Probabilistic Number Theory

A heuristic in analytic number theory stipulates that sets of positive integers cannot simultaneously be additively and multiplicatively structured. The practical verification of this heuristic is the source of a great number of difficult problems, including the well-known Hardy-Littlewood tuples conjecture. Conjectures of this type are also at least morally equivalent to the expectation that a \emph{generic} multiplicative function behaves randomly on additively structured sets. \\ In this thesis, we consider several problems involving the behaviour of multiplicative functions interacting with additively structured sets. First, we prove quantitative versions of mean value theorems due to Wirsing and Hal\'{a}sz for multiplicative functions that often take values outside of the unit disc, and provide some applications of this result to probabilistic number theory.\\ In a different direction, we refine a quantitative mean value theorem for multiplicative functions, and thereby significantly improve the existing upper bounds on the maximum size of partial sums of odd order non-principal Dirichlet characters, both unconditionally and assuming the Generalized Riemann Hypothesis, and show that our conditional results are nearly best possible unconditionally. \\ We also prove a quantitative bivariate Erd\H{o}s-Kac theorem on the joint distribution of pairs of values of certain additive functions, and use this probabilistic result to derive a weak version of Chowla's conjecture on binary correlations of the M\"{o}bius function, among other applications.\\ Solving a 60-year-old open problem of N.G. Chudakov, we show that a completely multiplicative function that only takes finitely many values, vanishes at only finitely many primes and whose partial sums are uniformly bounded, must be a non-principal Dirichlet character. We also solve a folklore conjecture due to Elliott, Ruzsa and others on the gaps between consecutive values of a unimodular completely multiplicative function, showing that these gaps cannot be uniformly large. This latter result is a corollary of several stronger results that are proved regarding the distribution of consecutive values of multiplicative functions. \\ Finally, we make some progress on some natural variants of Chowla's conjecture on sign patterns of the $\{+1,-1\}$-valued multiplicative functions. \\ Some of the aforementioned results are joint work with O. Klurman, or with Y. Lamzouri.Ph.D