论文标题
三元埃及分数带
Ternary Egyptian fractions with prime denominator
论文作者
论文摘要
对于Prime数字$ P $，令$ a_3（p）= | \ {m \ in \ mathbb {n}：\存在m_1，m_2，m_3 \ in \ mathbb {n}，\ frac {m} {p} = \ frac {1} {1} {m_1} {m_1} {m_1}+\ frac+\ frac {1} {1} {1} {m_2}+frac} | $。在2019年，卢卡（Luca）和帕帕拉迪（Pappalardi）证明了$ x（\ log x）^3 \ ll \ sum_ {p \ le x} a_ {3}（p）\ ll x（\ log x）^5 $。我们改进上限，显示$ \ sum_ {p \ le x} a_ {3}（p）\ ll x（\ log x）^3（\ log \ log \ log x）^2 $。
For a prime number $p$, let $A_3(p)= | \{ m \in \mathbb{N}: \exists m_1,m_2,m_3 \in \mathbb{N}, \frac{m}{p}=\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3} \} |$. In 2019 Luca and Pappalardi proved that $x (\log x)^3 \ll \sum_{p \le x} A_{3}(p) \ll x (\log x)^5$. We improve the upper bound, showing $\sum_{p \le x} A_{3}(p) \ll x (\log x)^3 (\log \log x)^2$.
