Title：Second Moment of Error Term for the Binary Egyptian Fractions

Speaker：Xuanxuan, Xiao

Time：2018.12.21   10:10-11:10

Venue：B1044

Abstract：Let $a$ be a fixed positive integer. For integer $n>0$, denote

$$R(n;a)={\rm card}\left\{(x,y)\in \mathbb{N}^2:\frac{a}{n}=\frac{1}{x}+\frac{1}{y}\right\},$$

and

$$S(x;a)=\sum_{\substack{n\leq x\\(n,a)=1}}R(n;a).$$

Then we have

$$S(x;a)=C_aN((\log N)^3+c_1(a)\log N+c_0(a))+\Delta(x;a).$$

We are interested in the size of the error term $\Delta(x;a)$ and get an asymptotic formula for its second moment, i.e.

$$\int_T^{2T}\Delta(x;a)^2dx=C_1'(a)T^{5/3}+O(T^{5/3-1/360}).$$