Topics in explicit number theory
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Date
2023
Authors
Hasanalizade, Elchin
University of Lethbridge. Faculty of Arts and Science
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Publisher
Lethbridge, Alta. : University of Lethbridge, Dept. of Mathematics and Computer Science
Abstract
This doctoral thesis is made up of a collection of papers of the author. All of these papers are already published. The structure of this thesis is as follows. After the introduction, there are four chapters.
In Chapters 2 and 3, we obtain explicit bounds for the number of non-trivial zeros of the Riemann zeta function and Dedekind zeta functions, which improve previous results of Trudgian. This improvement is based on ideas from previous works of Bennett et al., Kadiri and Ng and Trudgian. This is a joint work with Quanli Shen and Peng-Jie Wong.
In Chapter 4, we apply explicit results from transcendental number theory due to Bugeaud, Laurent and Matveev to completely solve a Diophantine inequality involving the Fibonacci numbers and to study a particular case of the Terai-Shinsho conjecture.
In Chapter 5, using elementary methods and explicit bounds for primes in arithmetic progressions due to Bennett et al. we study two Diophantine equations which involve multiplicative functions.
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Keywords
Number theory , Zeros of zeta functions , Linear forms in logarithms , Diophantine equations , Multiplicative functions