Lower Bound for Zeros in The Character Table of The Symmetric Group with an n-Core Index

For any two partitions $\lambda$ and $\mu$ of a positive integer $N$, let $\chi_{\lambda}(\mu)$ denote the value of the irreducible character of the symmetric group $S_{N}$ associated with $\lambda$, evaluated at the conjugacy class of elements whose cycle type is determined by $\mu$. The quantity $Z_{t}(N)$ is defined as $$ Z_{t}(N):= \#\{(\lambda,\mu): \chi_{\lambda}(\mu) = 0 \quad \text{with $\lambda$ a $t$-core}\}. $$ We establish the bound $$ \max\limits_{1\leq t \leq N} Z_{t}(N) \geq c_{t}(N)p(N-t)\geq \frac{2\pi p(N)^{2}}{1.01e\sqrt{6N}\log N} \biggl(1+O(N^{-\frac{1}{2}}\log N)\biggr), $$ where $p(N)$ denotes the number of partitions of $N$. Also, we give lower bounds for $Z_{t}(N)$ in different ranges of $t$ and obtain a lower bound for the total number of zeros in the character table of $S_N$.