A folklore conjecture on the partition function asserts that the density of odd values of $p(n)$ is $\frac{1}{2}$. In general, for a positive integer t, let $p_t(n)$ be the t-multipartition function and ${\delta }_t$ be the density of the odd values of $p_t(n)$. It is widely believed that ${\delta }_t$ exists. Given an odd integer a and an integer b depending on a and t, Judge and Zanello framed an infinite family of conjectural congruence relations on $p_t(an+b)(\mathrm{mod}2)$ which establishes a striking connection between ${\delta }_a$ and ${\delta }_1$. As a special case $t=1$, it implies that ${\delta }_1>0$ if $(3,a)=1$ and ${\delta }_a>0$. This conjecture was proved for several values of a by Judge, Keith and Zanello. In this paper we prove that the conjecture is true for $a={\ell }^{\alpha }$ is a prime power with $\ell \ge 5$ and $a=3$.

关于分割函数的民间传说猜想断言 $p（ñ）$ 是 $\frac{\mathrm{1个}}{\mathrm{2个}}$。通常，对于正整数t，令$p_{Ť}（ñ）$是t- multipartition函数，并且${\delta }_{Ť}$ 是的奇数值的密度 $p_{Ť}（ñ）$。人们普遍认为${\delta }_{Ť}$存在。给定一个取决于a和t的奇数整数a和一个整数b，Judge和Zanello构造了一个无限的猜想同余关系族$p_{Ť}（\mathrm{一个}ñ+b）（\mathrm{国防部}\mathrm{2个}）$ 在两者之间建立了惊人的联系 ${\delta }_{\mathrm{一个}}$ 和 ${\delta }_{\mathrm{1个}}$。作为特殊情况$Ť=\mathrm{1个}$，则表示 ${\delta }_{\mathrm{1个}}>0$ 如果 $（3，\mathrm{一个}）=\mathrm{1个}$ 和 ${\delta }_{\mathrm{一个}}>0$。法官Keith和Zanello对a的几个值证明了这一猜想。在本文中，我们证明该猜想是正确的$\mathrm{一个}={\ell }^{\alpha }$ 是一个主要力量 $\ell \ge 5$ 和 $\mathrm{一个}=3$。