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

### MSC:

 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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