Visibility of lattice points in high dimensional spaces and extreme values of combinations of Dirichlet L-functions

This thesis is divided into three major topics. In the first, we study questions concerning the distribution of lattice points in dimensions two and higher. We give asymptotic formulas for the number of integer lattice points of fixed index visible from certain admissible sets. We also study the shape of the body PA₁, \dots, PA_k when the lattice points A₁, \dots, A_k move inside a given large ball, and $P$ is a lattice point visible from all A_i's. In the second part, we study the phenomenon of similar ordering of Farey fractions and their generalizations. The notion of similar ordering for pairs of rationals was first introduced by Hardy, Littlewood and P\'olya. Later A.E. Mayer proved that pairs of Farey fractions in \mathcal{F}_Q are similarly ordered when $Q$ is large enough. We generalize Mayer's result to Ducci iterates of Farey sequences and visible points in convex regions. We also generalize the notion of index for Farey sequences and study the distribution of this generalization for large values of $Q$. In the third part of this thesis, we work with large values of certain Dirichlet series and their partial sums. In particular, we examine a `short' partial sum of the Riemann Zeta function, \zeta_N(s) = \sum_{n\le N}\frac{1}{n^s}. We obtain large values of \zeta_N\left(\frac{1}{2}+it\right) for $t\in[\sqrt{T}, T]$ when N\le (\log T)^\lambda with $0<\lambda<\sqrt{e}$. We also adapt the methods by K. Soundararajan \cite{Sound} and A. Bondarenko, K. Seip \cite{Bond} to obtain large values on the critical line for a certain linear combination of Dirichlet $L$-functions.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2018

Author dc:creator

• Tamazyan, Albert

Contributors dc:contributor

• Berndt, Bruce C.
• Zaharescu, Alexandru
• Boca, Florin
• Robles, Nicolas M

• Visible Points, Farey Sequences

Statement dc:rights

• Copyright 2018 Albert Tamazyan

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/101560

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/101560