2026-09-22T01:15:13Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/132565 2026-02-20 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2026-02-19 without embargo terms The student, Tanmay Khale, accepted the attached license on 2025-12-03 at 10:19. The student, Tanmay Khale, submitted this Dissertation for approval on 2025-12-03 at 12:54. This Dissertation was approved for publication on 2025-12-05 at 10:40. DSpace SAF Submission Ingestion Package generated from Vireo submission #23040 on 2026-02-19 at 18:26:32 Topics in analytic number theory Khale, Tanmay 2025-12-05 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 eng 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. 2025-12 Thesis https://hdl.handle.net/2142/132565 Copyright 2025 Tanmay Khale Mathematics Mathematics University of Illinois Urbana-Champaign Ph.D. Dissertation