Combinatorial methods in number theory: Sieve theory and special functions

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.

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

• Xie, Likun

Contributors dc:contributor

• Zaharescu, Alexandru
• Berndt, Bruce Carl
• Reznick, Bruce
• Nath, Kunjakanan

• Mctintosh conjecture
• almost primes
• vector sieves
• elliptic curves
• twin primes

Statement dc:rights

• Copyright 2025 Likun Xie

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/129422

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F129422