# Meeting Details

Title: "Analytic methods for Diophantine problems" Ph.D. Thesis Defense Jingjing Huang, Advisers: Winnie Li & Bob Vaughan, Penn State We are mainly concerned with the Diophantine equation $$\frac{a}{n}=\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_k}$$ and its number of positive integer solutions $R_k(n;a)$. We begin with the binary case $k=2$. Now the distribution of the function $R_2(n;a)$ is well understood. More precisely, by averaging over $n$, the first moment and second moment behaviors of $R_2(n;a)$ have been established. For instance, one of our results is $$\sum_{\substack{n\le N\\(n,a)=1}}R_2(n;a)=N P_2(\log N;a)+O_a(N\log^5 N),$$ where $P_2(\cdot;a)$ is a quadratic function whose coefficients depend on $a$. Furthermore, we have shown that, after normalisation, $R_2(n;a)$ satisfies Gaussian distribution, which is an analog of the classical theorem of Erd\H{o}s and Kac, $$\lim_{N\to\infty}\frac1N\rm{card}\left\{n\le N:\frac{\log R_2(n;a)-(\log 3)\log\log n}{(\log 3)\sqrt{\log\log n}}\le z\right\}=\frac1{\sqrt{2\pi}}\int_{-\infty}^{z}e^{-\frac{t^2}{2}}dt.$$ On the other hand, we change the point of view and study the set of exceptional numbers" that do not possess binary representations. Let $E_a(N)$ denote the number of $n\le N$ such that $R_2(n;a)=0$. It is established that when $a\ge3$ we have $$E_a(N)\sim C(a) \frac{N(\log\log N)^{2^{m-1}-1}}{(\log N)^{1-1/2^m}},$$ with $m$ defined in the talk. I will explain how to prove this theorem. The next project would be to study the ternary case $k=3$. While the conjecture, by Erd\H{o}s, Straus and Schinzel, that for fixed $a\ge 4$, we have $R_3(n;a)>0$ when $n$ is sufficiently large, is still wide open, here I will talk about some partial results on the mean value $\sum_{n\le N}R_3(n;a)$ if time permits.