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

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