Equidistribution and inequalities for partitions into powers

Alexandru Ciolan

Published 2020-02-13Version 1

In this paper, we study partitions into powers with an odd and with an even number of parts. We show that the two quantities are equidistributed, and that the one which is bigger alternates according to the parity of $n.$ This extends a similar result established by the author (2020) for partitions into squares. By modifying a certain argument from the proof in the case of partitions into squares and by invoking a bound on Gauss sums found by Banks and Shparlinski (2015) using the work of Cohn and Elkies (2003) on lower bounds for the center density in the sphere packing problem, we generalize this result to partitions into higher powers.