Asymptotics of Character Sums

Overview

Objectives/Goals In this project, we aim to prove certain properties about a particular function c(n) = b_nr(n). This is where b_n is a Boolean function with b_n being 1 if n = x^2 + y^2 for some integers x and y or 0 otherwise and r_chi(n) is the sum of all of the Dirichlet characters of d, where d divides n. The function c(n) sums the all of the chi values of the divisors of a certain number n if and only if n can be expressed as the sum of two squares. Therefore, the question we ask is the following: What are the asymptotics of the character sums of the function c(n)? Results In order to investigate this problem, we first represent the character sum of r(n) as an asymptotic and prove that the asymptotic is roughly L(1, chi) with a small error term. Additionally, we compute a representation for the character sum c(n) as an Euler product, and also find error bounds on the asymptotic for the character sum. Conclusions/Discussion We analyzed the asymptotic, or growth rate, of a very special function c(n) which describes a very particular group of primes. In specific, our growth rate describes the group of primes which are dependent on two character values. Additionally, we found some error bounds on how accurate our asymptotic is.

Awards (1)

• Second Place

Competition history

• CSEF 2018 Mathematical Sciences · Entry S1510

Resources

• CSEF project summary ↗

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