Geodesic
ON SOME QUESTIONS OF PARTITIO NUMERORUM: TRES CUBI
Glasgow mathematical journal, Tome 63 (2021) no. 1, pp. 223-244

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DOI

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 $${\sum_{\substack{ n\le x\\ n\equiv a\,\mathrm{mod}\,q }} r_3(n).\}$$ 
VAUGHAN, R. C. ON SOME QUESTIONS OF PARTITIO NUMERORUM: TRES CUBI. Glasgow mathematical journal, Tome 63 (2021) no. 1, pp. 223-244. doi: 10.1017/S0017089520000142
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     author = {VAUGHAN, R. C.},
     title = {ON {SOME} {QUESTIONS} {OF} {PARTITIO} {NUMERORUM:} {TRES} {CUBI}},
     journal = {Glasgow mathematical journal},
     pages = {223--244},
     year = {2021},
     volume = {63},
     number = {1},
     doi = {10.1017/S0017089520000142},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089520000142/}
}
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