Representation as sum of $k$th powers of a residue class

Amita Malik (University of Illinois at Urbana-Champaign)

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 provided a ``simpler" asymptotic formula in the case $k=2$. In this talk, we discuss partitions into parts from a specific set $\mathcal{A}_k(a,b) :=\left\{ m^k : m \in \mathbb{N}, m \equiv a \pmod{b} \right\}$, for fixed positive integers $k$, $a,$ and $b$. Using the circle method, we give an asymptotic formula for the number of such partitions, thus generalizing the results of Wright, and Vaughan. We also discuss the parity problem for such partitions which gives O.~Kolberg's renowned theorem in the case of the ordinary partition function. This is joint work with Bruce~Berndt and Alexandru~Zaharescu.

Tuesday, November 29, at 4:00pm in HBH 227

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