Zeros in the character tables of symmetric groups with an -core index

Let $\mathcal{C}_n =\left [\chi_{\lambda}(\mu)\right]_{\lambda, \mu}$ be the character table for $S_n,$ where indices $\lambda$ and $\mu$ run over $p(n)$ many integer partitions of $n.$ In this note we study $Z_{\ell}(n),$ number zero entries $\chi_{\lambda}(\mu)$ in $\mathcal{C}_n,$ is an $\ell$-core partition For every prime $\ell\geq 5,$ prove asymptotic formula form $$ Z_{\ell}(n)\sim \alpha_{\ell}\cdot \sigma_{\ell}(n+\delta_{\ell})p(n)\gg_{\ell} n^{\frac{\ell-5}{2}}e^{\pi\sqrt{2n/3}},$$ $\sigma_{\ell}(n)$ a twisted Legendre symbol divisor function, $\delta_{\ell}:=(\ell^2-1)/24,$ $1/\alpha_{\ell}>0$ normalization Dirichlet $L$-value $L\left(\left(\frac{\cdot}{\ell}\right),\frac{\ell-1}{2}\right).$ primes $\ell$ $n>\ell^6/24,$ show that $\chi_{\lambda}(\mu)=0$ whenever are both $\ell$-cores. Furthermore, if $Z^*_{\ell}(n)$ indexed by two $\ell$-cores, then 5$ obtain Z^*_{\ell}(n)\sim \alpha_{\ell}^2 \cdot \sigma_{\ell}( n+\delta_{\ell})^2 \gg_{\ell} n^{\ell-3}.