# Departmental PhD Thesis Exam

Event Information Topics in Multiplicative and Probabilistic Number Theory
15:10 on Wednesday February 14, 2018
16:00 on Wednesday February 14, 2018
BA6183, Bahen Center, 40 St. George St.
Alexander Mangerel
University of Toronto

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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, e.g., the Goldbach conjecture. Relatedly, we expect that a multiplicative function, unless it has a special form, 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 mean value theorems for unbounded multiplicative functions, extending work of Wirsing and Hal\'{a}sz, who treated the bounded case. As a consequence, we prove a multivariate Poisson limit theorem for vectors of additive functions.

Next, we consider the particular collection of periodic, completely multiplicative functions, also known as Dirichlet characters. We significantly improve the existing upper bounds on the maximum size of partial sums of odd order Dirichlet characters, both unconditionally and assuming the Generalized Riemann Hypothesis. We also show that our conditional results are nearly best possible unconditionally.

We furthermore prove that non-principal Dirichlet characters are the only completely multiplicative functions that: i) only take finitely many values, ii) vanish at only finitely many primes and iii) whose partial sums are uniformly bounded. This solves a 60-year-old open problem of N.G.

Thirdly, we prove a quantitative bivariate Central Limit Theorem for pairs of values of certain additive functions, finding a quantitative error term in this approximation. We use this probabilistic result to prove a partial result in the direction of Chowla's conjecture on two-point correlations of the M\"{o}bius function.

Chudakov. We also solve a folklore conjecture due to Elliott, Ruzsa and others, showing that the gaps between consecutive values of a unimodular completely multiplicative function cannot be uniformly large. We prove several stronger results in this direction. For instance, answering a question of K\’{a}tai, we classify the set of all unimodular completely multiplicative functions $f$ such that $\{f(n)\}_n$ is dense in $\mathbb{T}$ and for which the sequence of pairs $(f(n),f(n+1))$ is dense in $\mathbb{T}^2$.

Finally, we make some progress on some natural variants of Chowla's conjecture on sign patterns of the Liouville function. In particular, we prove that certain natural collections of multiplicative functions $f: \mathbb{N} \rightarrow \{-1,+1\}$ are such that the tuples of values they produce on \emph{almost all} 3- and 4-term arithmetic progressions equidistribute among all sign patterns of length 3 and 4, respectively.

Some of the aforementioned results are joint work with O. Klurman, or with Y. Lamzouri.

Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.