University of Illinois Urbana-Champaign

Effective estimates for sums of Kloosterman sums, modular forms, and applications

Gonzalez-Pagan, Oscar E

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https://hdl.handle.net/2142/120190

Description

Title
Effective estimates for sums of Kloosterman sums, modular forms, and applications
Author(s)
Gonzalez-Pagan, Oscar E
Issue Date
2023-01-05
Director of Research (if dissertation) or Advisor (if thesis)
Ahlgren, Scott
Doctoral Committee Chair(s)
  • Ahlgren, Scott
  • Ford, Kevin
Committee Member(s)
  • Thorner, Jesse
  • Zaharescu, Alexandru
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
  • Number Theory
  • Modular Forms
  • Kloosterman Sums
Abstract
In this thesis we explore applications of effective estimates for sums of Kloosterman sums, and consider some problems related to modular forms. In Chapter 1 we give a substantial improvement for the error term in the asymptotic formula for the smallest parts function spt(n) of Andrews. Our methods depend on an explicit bound for sums of Kloosterman sums of half integral weight. In the next chapter we establish an asymptotic formula with an effective bound on the error term for traces of singular moduli. The third chapter establishes the irreducibility of a family of polynomials associated to the zeros of Eisenstein series on SL2(Z). The final chapter provides a result about the factorization of the determinants of certain Hankel matrices related to Eisenstein series.
Graduation Semester
2023-05
Type of Resource
Thesis
Copyright and License Information
Copyright 2023 Oscar Gonzalez-Pagan

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