On the representation of integers as sums of triangular numbers. (English) Zbl 0828.11057

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 \(n \leq (4R^2 - k)/8\).


11P82 Analytic theory of partitions
11F11 Holomorphic modular forms of integral weight
11P21 Lattice points in specified regions
11H06 Lattices and convex bodies (number-theoretic aspects)
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