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

THE RIEMANN ZETA FUNCTION AS AN EQUIVARIANT DIRAC INDEX

In this note an interpretation of Riemann's zeta function is provided in terms of an ℝ-equivariant L2-index of a Dirac–Ramond type operator, akin to the one on (mean zero) loops in flat space constructed by the present author and T. Wurzbacher. We build on the formal similarity between Euler's partitio numerorum function (the S1-equivariant L2-index of the loop space Dirac–Ramond operator) and Riemann's zeta function. Also, a Lefschetz–Atiyah–Bott interpretation of the result together with a generalization to M. Lapidus' fractal membranes are also discussed. A fermionic Bost–Connes type statistical mechanical model is presented as well, exhibiting a "phase transition at (inverse) temperature β = 1", which also holds for some "well-behaved" g-prime systems in the sense of Hilberdink–Lapidus.

Some problems in partitio numerorum

We consider some unconventional partition problems in which the parts of the partition are restricted by divisibility conditions, for example, partitions n = a1+…+ak into positive integers a1, …, ak such that a1 ∣ a2 ∣ … ∣ ak. Some rather weak estimates for the various partition functions are obtained.