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

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Gonzalez-Pagan, Oscar E.

Contributors dc:contributor

• Ahlgren, Scott
• Ford, Kevin
• Thorner, Jesse
• Zaharescu, Alexandru

• Number Theory
• Modular Forms
• Kloosterman Sums

Statement dc:rights

• Copyright 2023 Oscar Gonzalez-Pagan

Language dc:language

en, eng

Handle dc:identifier

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

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F120190