Topics in analytic number theory

Khale, Tanmay

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

Description

Title

Topics in analytic number theory

Author(s)

Khale, Tanmay

Issue Date

2025-12-05

Director of Research (if dissertation) or Advisor (if thesis)

• Ford, Kevin B
• Thorner, Jesse A

Doctoral Committee Chair(s)

Zaharescu, Alexandru

Committee Member(s)

Gabdullin, Mikhail

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• 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

Abstract

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.

Graduation Semester

2025-12

Type of Resource

Thesis

Handle URL

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

Copyright and License Information

Copyright 2025 Tanmay Khale

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