2026-09-20T09:58:36Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/129422 2025-10-20 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 2025-10-19 without embargo terms The student, Likun Xie, accepted the attached license on 2025-04-22 at 11:09. The student, Likun Xie, submitted this Dissertation for approval on 2025-04-22 at 11:28. This Dissertation was approved for publication on 2025-04-24 at 07:02. DSpace SAF Submission Ingestion Package generated from Vireo submission #21886 on 2025-10-19 at 18:18:36 Combinatorial methods in number theory: Sieve theory and special functions Xie, Likun 2025-04-24 Zaharescu, Alexandru Berndt, Bruce Carl Reznick, Bruce Nath, Kunjakanan Mctintosh conjecture almost primes vector sieves elliptic curves twin primes eng This dissertation explores various problems in analytic number theory, using a blend of sieve-theoretic methods, algebraic, and combinatorial techniques. The work is split into two main parts. First, we study sieve methods and applications. We investigate counting results for primes and almost primes with additional constraints, such as those related to orders of elliptic curves modulo prime powers, primes p with large power factors in p - b, and primes of the form p = 1 + m^2 + n^2 such that p + 2 is an almost prime. These problems are approached using vector sieve, linear sieve, semi-linear sieve, and other advanced sieve frameworks, often combined with analytic estimates like the Bombieri–Vinogradov theorem in various contexts. Second, we address Franel integrals and arithmetic properties. We prove McIntosh’s conjecture on certain multidimensional Franel integrals involving Bernoulli polynomials and higher-dimensional analogs. By unifying these two directions—sieve applications and Franel-type integrals—we illustrate the power of combinatorial-analytic techniques in tackling problems within number theory. In particular, we show how seemingly disparate areas (twin primes and almost primes, the orders of elliptic curves over finite fields, and integrals involving Bernoulli functions) can reinforce one another. 2025-05 Thesis https://hdl.handle.net/2142/129422 Copyright 2025 Likun Xie Mathematics Mathematics University of Illinois Urbana-Champaign Ph.D. Dissertation