Alexandru Ciolan: Equidistribution and inequalities for partitions into powers, 409-431

Abstract:

If $p_k(a,m;n)$ denotes the number of partitions of $n$ into $k$th powers with a number of parts that is congruent to $a$ modulo $m,$ recent work of the author (2020) showed that $p_2(0,2;n)\sim p_2(1,2;n)$ and that the sign of the difference $p_2(0,2;n)- p_2(1,2;n)$ alternates with the parity of $n$ as $n\to\infty.$ The aim of this paper is to study this problem in its full generality. By an analytic argument using the circle method and an upper bound on exponential Gauss sums related to center density estimates arising from the sphere packing problem, we prove that the same results hold for any $k\ge2.$ In addition, by a purely combinatorial argument, we show that the sign of the difference $p_k(0,2;n)- p_k(1,2;n)$ alternates with the parity of $n$ for a larger class of partitions.

Key Words: Asymptotics, circle method, equidistribution, Gauss sums, partition inequalities, power partitions, saddle-point method, sphere packing problem.

2020 Mathematics Subject Classification: Primary 11P82; Secondary 11P55, 11P83.

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