ON SOME QUESTIONS OF PARTITIO NUMERORUM: TRES CUBI

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 $${\sum_{\substack{ n\le x\\ n\equiv a\,\mathrm{mod}\,q }} r_3(n).\}$$