2026-09-19T03:23:37Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/120190 2023-09-04 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Ahlgren, Scott Ahlgren, Scott Ford, Kevin Thorner, Jesse Zaharescu, Alexandru 2023-05 application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2023-09-01 without embargo terms The student, Oscar Gonzalez-Pagan, accepted the attached license on 2022-12-21 at 09:47. The student, Oscar Gonzalez-Pagan, submitted this Dissertation for approval on 2022-12-21 at 10:29. This Dissertation was approved for publication on 2023-01-05 at 10:11. DSpace SAF Submission Ingestion Package generated from Vireo submission #18801 on 2023-09-01 at 17:06:47 Effective estimates for sums of Kloosterman sums, modular forms, and applications Gonzalez-Pagan, Oscar E. 2023-01-05 Number Theory Modular Forms Kloosterman Sums 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. Thesis eng https://hdl.handle.net/2142/120190 Copyright 2023 Oscar Gonzalez-Pagan Ph.D. Dissertation Mathematics University of Illinois at Urbana-Champaign Mathematics