Asymptotic formulas for general colored partition functions

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 $s_i (1le ile k)$, 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.