## Departmental PhD Thesis Exam – Alexander Mangerel

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

Wednesday, February 14, 2018
2:10 p.m.
BA6183

PhD Candidate:  Alexander Mangerel
Supervisor:  John Friedlander
Thesis title: Topics in Multiplicative and Probabilistic Number Theory

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Abstract:

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. An example of this is the well-known Hardy-Littlewood tuples conjecture, which asserts that, infinitely often, one should be able to find additive patterns of fairly general shape in the primes. Conjectures of this type are also at least morally equivalent to the expectation 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. Two main topics are studied: i) the estimation of \emph{mean values} of multiplicative functions, i.e., the limiting average behaviour of partial sums of multiplicative functions along an interval whose length tends to infinity; and ii) the estimation of \emph{correlations} of multiplicative functions, i.e., the behaviour of simultaneous values of multiplicative functions at arguments that are additively related. A number of applications of the study of these topics are also addressed.

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. This has a broad realm of applications. In particular we are able to extend a further theorem of Hal\'{a}sz, proving local limit theorems for vectors of certain types of additive functions. We thus confirm a probabilistic heuristic in the \emph{small deviation} regime and beyond for the functions in question.

In a different direction, we consider the collection of periodic, completely multiplicative functions, also known as Dirichlet characters. Upper bounds for the maximum size of the partial sums of these functions on intervals of positive integers is connected with the class number problem in algebraic number theory, and with I.M. Vinogradov’s conjecture on the distribution of quadratic non-residues. By refining a quantitative mean value theorem for multiplicative functions, 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 best possible unconditionally, up to a bounded power of $\log\log\log\log q$.

Regarding correlations, we prove a quantitative version of the bivariate Erd\H{o}s-Kac theorem. That is, we show that the joint distribution of pairs of values of certain additive functions is asymptotically an uncorrelated bivariate Gaussian, and find a quantitative error term in this approximation. We use this probabilistic result to prove a theorem on the joint distribution of certain natural variants of the M\”{o}bius function at additively-related integers as a partial result in the direction of Chowla’s conjecture on two-point correlations of the M\”{o}bius function. We also apply our result to understanding the set of pairs of consecutive integers with the same number of divisors.

A major theme in the thesis relates to how a multiplicative function can be rigidly characterized globally by certain local properties. As a first example, 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. This solves a 60-year-old open problem of N.G. Chudakov. 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 is a corollary of several stronger results that are proved regarding the distribution of consecutive values of multiplicative functions. For instance, we classify the set of all unimodular completely multiplicative functions $f$ such that $\{f(n)\}_n$ is dense in $\mb{T}$ and for which the sequence of pairs $(f(n),f(n+1))$ is dense in $\mb{T}^2$. In so doing, we resolve a conjecture of K\'{a}tai.

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 $f: \mb{N} \ra \{-1,+1\}$ 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.

A copy of the thesis can be found here: APMangerPhDThesisFeb13