Asymptotic formulas for general colored partition functions

TL;DR

This paper derives explicit asymptotic formulas with error bounds for general colored partition functions, extending classical results and recent work on related functions.

Contribution

It provides the first explicit asymptotic formula with error term for a broad class of colored partition functions.

Findings

Derived asymptotic formulas for general colored partition functions

Extended classical partition asymptotics to colored cases

Provided explicit error bounds in the formulas

Abstract

In 1917, Hardy and Ramanujan obtained the asymptotic formula for the classical partition function $p(n)$. The classical partition function $p(n)$ has been extensively studied. Recently, Luca and Ralaivaosaona obtained the asymptotic formula for the square-root function. Many mathematicians have paid much attention to congruences on some special colored partition functions. In this paper, we investigate the general colored partition functions. Given positive integers $1=s_1<s_2<\dots <s_k$ and $\ell_1, \ell_2,\dots , \ell_k$. Let $g(\mathbf{s}, \mathbf{l}, n)$ be the number of $\ell$-colored partitions of $n$ with $\ell_i$ of the colors appearing only in multiplies of $s_i\ (1\le i\le k)$, where $\ell = \ell_1+\cdots +\ell_k$. By using the elementary method we obtain an asymptotic formula for the partition function $g(\mathbf{s}, \mathbf{l}, n)$ with an explicit error term.