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

Abstract

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 $PA_1, \dots, PA_k$ when the lattice points $A_1, \dots, A_k$ 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