Colin Defant

Mentor: Dr. Krishna Alladi
College of Liberal Arts and Sciences
 
"When I was a freshman, I started proving results about certain sequences and wrote a paper about them. I then attended the REU in Algebra and Discrete Mathematics at Aurburn University and wrote several other original research papers. I found that I greatly enjoy proving new theorems and writing papers. Dr. Bona told me about the University Scholars Program, and I was immediately interested because I wanted the chance to work with him."

Major

Mathematics

Minor

N/A

Research Interests

  • Number Theory
  • Combinatorics
  • Dynamical Systems

Academic Awards

  • University Scholars Program, 2015, 2016
  • National Merit Scholarship Finalist
  • Mu Alpha Theta National Mathematics Honor Society Scholarship
  • UF Presidential Platinum Scholarship
  • Barry Goldwater Excellence in Education Award

Organizations

  • President of University Math Society
  • Director of UF Problem Solving Group

Volunteer

  • Director, Writer, and Editor for the Math Majors of America Tournament for High Schools at UF
  • Director of the UF Integration Bee
  • Volunteering with Children at the Temple Terrace Library

Hobbies and Interests

  • Mathematics
  • Composing Electronic Music and Rapping
  • Pole Vaulting
  • Crafts

Research Description

Small Divisors of Squarefree Integers
Between 1987 and 1989, Alladi, Erdos, and Vaaler wrote two articles that provided concrete support for the heuristic idea that "large divisors of a squarefree integer have more prime divisors than the small ones." More precisely, they investigated the ratio between the quantities sum_{d|n}(h(d)) and sum_{d|n, d<= n^{1/(t+1)}}(h(d)) when t>=1, n is a squarefree positive integer, and h:N-->R is a multiplicative arithmetic function (meaning h(lm)=h(l)h(m) whenever gcd(l,m)=1) that satisfies 0=1 is a rational number with continued fraction expansion [a_0,a_1,...,a_r], then R(h,n,t)<=1+a_0+a_1+...+a_r. In particular, R(h,n,t)<=t+1 if t is an integer. Soundararajan also showed that R(h,n,t)<=10*sqrt{omega(n)} if t is irrational. These upper bounds for R(h,n,t) beckon for improvement. First, Alladi conjectures that if t is an integer, then R(h,n,t)<=4+o(1), where o(1) tends to 0 as omega(n)-->infinity. Hence, my primary goal for this research project will be to find some absolute constant which serves as an upper bound for R(h,n,t) (possibly only when t is an integer). Even if this proves too difficult a task, it still seems possible to obtain an upper bound for R(h,n,t) that does not depend on n when t is irrational. Indeed, we know that R(h,n,t)<=t+1 when t is an integer; one should be able to produce similar bounds that do not depend on the membership of t in Z or Q.