On the representation of integers as sums of triangular numbers (Q1897770)

This survey article treats \(\delta_k (n)\), the number of representations of a positive integer \(n\) as the sum of \(k\) triangular numbers. Using the theory of modular forms the authors calculate \(\delta_k (n)\) explicitly for \(k = 2\), 3, 4, 6, 8, 10, 12 and 24. The results for \(k = 24\) reveal connections with 24-dimensional Leech lattices and with Ramanujan's function \(\tau (n)\). It is also shown that the number of lattice points in a \(k\)-dimensional sphere of radius \(R\) centered at \(({1 \over 2}, {1 \over 2}, \ldots, {1 \over 2})\) is \(2^k\) times the sum \(\sum \delta_k (n)\) extended over all \(n \leq (4R^2 - k)/8\).