An asymptotic for the representation of integers as sums of triangular numbers

Abstract

Motivated by the result of Rankin for representations of integers as sums of squares, we use a decomposition of a modular form into a particular Eisenstein series and a cusp form to show that the number of ways of representing a positive integer $n$ as the sum of $k$ triangular numbers is asymptotically equivalent to the modified divisor function ${\sigma }_{2k-1}^{♯}\left(2n+k\right)$.