Power partitions

In 1918, Hardy and Ramanujan published a seminal paper which includes an asymptotic formula for the partition function. They claim without proof an asymptotic equivalence for pk(n)pk(n), the number of partitions of a number n into k  -th powers. In this paper, we establish an asymptotic formula for pk(n)pk(n), using the Hardy–Littlewood circle method. We also derive a formula for the difference function pk(n+1)−pk(n)pk(n+1)−pk(n). As a necessary step in the proof, we obtain a non-trivial bound on exponential sums of the form ∑m=1qe(amkq).