Distribution of sequences related to L-functions
This thesis consists of three projects. The first project focuses on the distribution of zeros of linear combinations of derivatives of $L$-functions. We consider a collection of such combinations and prove asymptotic formulas for the supremum of the real parts of their zeros. Moreover, an investigation of an inverse-type question related to the case of the Riemann zeta function is included. In the second part of this thesis, we expand the class of Dirichlet series whose monotonicity properties are known. In particular, we describe a large class of Dirichlet series that are not logarithmically completely monotonic. Using similar techniques, an equivalent formulation of the Riemann Hypothesis for the Ramanujan-tau $L$-function is provided. The last project is related to walks to infinity. Our main object is the subset $P$ of the complex plane that includes all the primes of all rings of integers of all imaginary quadratic fields. One would want to know if it is possible to walk to infinity stepping only on points in $P$ and such that the sequence of lengths of steps used in the process is bounded. However, the problem is surprisingly connected to some famous and notoriously difficult unsolved problems. We study more general walks on the set $P$, where the length of the steps is not forced to be bounded throughout the walk.
riemann zeta, L-functions, distribution, zeros, monotonicity, walks
University of Illinois - Urbana-Champaign
Networked Digital Library of Theses and Dissertations
University of Illinois at Urbana-Champaign