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ON SOME QUESTIONS OF PARTITIO NUMERORUM: TRES CUBI

Part of: Additive number theory; partitions Multiplicative number theory

Published online by Cambridge University Press:  21 April 2020

R. C. VAUGHAN
R. C. VAUGHAN*
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA16802, USA, e-mail: rcv4@psu.edu
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Abstract

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

MSC classification

Primary: 11P05: Waring's problem and variants
Secondary: 11P21: Lattice points in specified regions 11P32: Goldbach-type theorems; other additive questions involving primes 11N36: Applications of sieve methods
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 1 , January 2021 , pp. 223 - 244
Copyright
© The Author(s) 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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Footnotes

In Memoriam Christopher Hooley FRS, 1928–2018

References

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