On the asymptotics of certain colored partitions
Lukas Mauth
TL;DR
The paper proves an asymptotic formula for the logarithm of two-colored partition counts $a_r(n)$, defined by the generating function $G_r(q)=P(q)P(q^r)$. Using the Hardy–Ramanujan Circle Method and modular transformation laws for eta-quotients, it derives a Rademacher-type asymptotic with explicit constants that hold for all positive integers $r$, confirming Guadalupe's conjecture for primes and extending it to general $r$. The leading term is $\log(a_r(n)) \sim \pi \sqrt{\frac{2n(1+r^{-1})}{3}}$, accompanied by a $-\frac{5}{4}\log(n)$ correction, a constant term, and a $-\frac{c_r}{24\sqrt{6n}}$ refinement, where $c_r=\frac{135}{\pi\sqrt{1+r^{-1}}}+\pi\sqrt{\frac{(1+r)^3}{r}}$. The work highlights the power of circle-method techniques for eta-quotients beyond small primes and unifies several prior results into a general asymptotic for two-colored partitions.
Abstract
We will prove an infinite family of asymptotic formulas for the logarithm of certain two-colored partitions. An infinite sub-family of these asymptotics was posed as a conjecture by Guadalupe.