Combinatorial methods in number theory: Sieve theory and special functions

Xie, Likun

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

Description

Title

Combinatorial methods in number theory: Sieve theory and special functions

Author(s)

Xie, Likun

Issue Date

2025-04-24

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

• Zaharescu, Alexandru
• Berndt, Bruce Carl

Doctoral Committee Chair(s)

Reznick, Bruce

Committee Member(s)

Nath, Kunjakanan

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Abstract

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.

Graduation Semester

2025-05

Type of Resource

Thesis

Handle URL

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

Copyright and License Information

Copyright 2025 Likun Xie

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