On a Variance Associated with the Distribution of Real Sequences in Arithmetic Progressions
Open Access
Author:
Ding, Pengyong
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
February 02, 2021
Committee Members:
Robert Charles Vaughan, Dissertation Advisor/Co-Advisor Robert Charles Vaughan, Committee Chair/Co-Chair Wen-Ching Winnie Li, Committee Member Ae Ja Yee, Committee Member Mary Kathleen Heid, Outside Member Mark Levi, Program Head/Chair
Keywords:
variance distribution sequences asymptotics residue classes circle method
Abstract:
This dissertation is composed of two parts. The first part concerns the general result on the following variance associated with the distribution of a real sequence $\{a_n\}$ in arithmetic progressions:
\begin{equation*}
V(x,Q)=\sum_{q\le Q}\sum^q_{a=1}|A(x;q,a)-f(q,a)M(x)|^2,
\end{equation*}
where $A(x;q,a)$ represents the sum of $\{a_n\}$ in the residue class of $a$ modulo $q$, and $f(q,a)$ and $M(x)$ approximately reflect the local and global properties of $\{a_n\}$ respectively. We will give a brief history of the problem and introduce two powerful methods, and then we will provide the standard initial procedure.
The second part is an example on calculating the variance $V(x,Q)$ when $a_n=r_3(n)$, the number of (ordered) representations of $n$ as the sum of three positive cubes:
\begin{equation*}
r_3(n)=\displaystyle\sum_{\substack{
x_1, x_2, x_3 \\
x_1^3+x_2^3+x_3^3=n
}} 1.
\end{equation*}
We will introduce the properties of the function and show how to calculate the main terms and estimate the error terms. The conclusion will be stated as Theorem 5.1. Finally, several special cases and similar questions will be listed at the end of the dissertation.
