On the counting function of sets with even partition functions

TL;DR

This paper investigates the growth rate of the counting function of specific sets defined via partition functions with even partition functions, providing improved bounds on their asymptotic behavior.

Contribution

It improves the constant in the asymptotic upper bound for the counting function of sets with even partition functions, refining previous results.

Findings

Enhanced the constant c(q) in the asymptotic bound

Provided sharper estimates for the counting function

Extended understanding of partition functions modulo 2

Abstract

Let q be an odd positive integer and P \in F2[z] be of order q and such that P(0) = 1. We denote by A = A(P) the unique set of positive integers satisfying \sum_{n=0}^\infty p(A, n) z^n \equiv P(z) (mod 2), where p(A,n) is the number of partitions of n with parts in A. In [5], it is proved that if A(P, x) is the counting function of the set A(P) then A(P, x) << x(log x)^{-r/\phi(q)}, where r is the order of 2 modulo q and \phi is Euler's function. In this paper, we improve on the constant c=c(q) for which A(P,x) << x(log x)^{-c}.