Topics in analytic number theory

This thesis, consisting of two chapters, proves several new theorems concerning L-functions and the distribution of primes. In the first chapter, we establish the first explicit form of the Vinogradov–Korobov zero-free region for Dirichlet L-functions. In the second chapter, we generalize recent work on large gaps between primes to imaginary quadratic fields. Suppose K is an imaginary quadratic field, and let N_K denote the field norm on O_K. For x₀ in O_K and r > 0, let (x₀, r) = { x in O_K : |N_K(x − x₀)| < r }. Define G_K(X) = max { r > 0 : there exists x₀ in O_K such that |N_K(x₀)| ≤ X and B(x₀, r) contains no primes }. We show that G_K(X) is at least c_K (log X) (log₂ X · log₄ X) / log₃ X for some constant c_K > 0 depending only on K.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• Khale, Tanmay

Contributors dc:contributor

• Ford, Kevin B
• Thorner, Jesse A
• Zaharescu, Alexandru
• Gabdullin, Mikhail

• analytic number theory
• prime number theory
• Dirichlet L-functions
• L-functions
• zero-free regions
• primes in arithmetic progressions
• prime gaps
• bounded gaps between primes
• large gaps between primes
• prime k-tuples
• Gaussian integers
• Gaussian primes

Statement dc:rights

• Copyright 2025 Tanmay Khale

Language dc:language

en

Handle dc:identifier

https://hdl.handle.net/2142/132565

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/132565