This paper is concerned with the function r3(n), the number of representations of n as the sum of at most three positive cubes, $$r_3(n) = {\mathrm{card}}\{\mathbf m\in\mathbb Z^3: m_1^3+m_2^3+m_3^3=n, m_j\ge1\}.$$ , Our understanding of this function is surprisingly poor, and we examine various averages of it. In particular $${\sum_{m=1}^nr_3(m),\,\sum_{m=1}^nr_3(m)^2}$$ and $${\sum_{\substack{ n\le x\\ n\equiv a\,\mathrm{mod}\,q }} r_3(n).\}$$

中文翻译：

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关于 PARTITIO NUMERORUM 的一些问题：TRES CUBI

本文关注函数r3(n)，表示的数量n作为最多三个正立方的总和，$$r_3(n) = {\mathrm{card}}\{\mathbf m\in\mathbb Z^3: m_1^3+m_2^3+m_3^3=n, m_j\ge1\}.$$, 我们对这个函数的理解出奇的差，我们检查了它的各种平均值。特别是$${\sum_{m=1}^nr_3(m),\,\sum_{m=1}^nr_3(m)^2}$$和$${\sum_{\substack{ n\le x\\ n\equiv a\,\mathrm{mod}\,q }} r_3(n).\}$$