Amita Malik

Affiliation: University of Illinois


Title Of Talk: Partitions into $k$th powers of terms in an arithmetic progression

Abstract: G. H. Hardy and S. Ramanujan established an asymptotic formula for the number of unrestricted partitions of a positive integer, and claimed a similar asymptotic formula for the number of partitions into perfect $k$th powers, which was later proved by E. M. Wright. Recently, R. C. Vaughan gave a simpler asymptotic formula in the case $k=2$. We study partitions into parts from a specific set $A_k(a_0,b_0) :=\left\{ m^k : m \in \mathbb{N} , m\equiv a_0 \pmod{b_0} \right\}$, for fixed positive integers $k$, $a_0,$ and $b_0$, and obtain an asymptotic formula for the number of such partitions. This is a generalization of the earlier mentioned results by Wright and Vaughan. We also discuss the parity of the number of these partitions. This is joint work with Bruce Berndt and Alexandru Zaharescu.

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