Average divisibility in character tables of $\mathrm{GL}_2(\mathbb{F}_q)$

Let $q$ range over odd prime powers and let $G_q=\mathrm{GL}_2(\mathbb{F}_q)$.

Fix a prime number $\ell$.

Motivated by work of Peluse and Soundararajan on Miller's conjecture for character tables of symmetric groups, we study the proportion of entries in the character table of $G_q$ which are not divisible by $\ell$, in the sense of divisibility in the ring of algebraic integers.

We prove that $N_\ell(q)=\frac{q^4}{2}+O_\epsilon(q^{3+\epsilon})$ for every $\epsilon>0$, where $N_\ell(q)$ denotes the number of entries which are not divisible by $\ell$.

We also show that the number of zero entries is $\frac{q^4}{2}+O_\epsilon(q^{3+\epsilon})$.

Consequently, the proportion of all entries not divisible by $\ell$ tends to $1/2$, while the proportion of nonzero entries not divisible by $\ell$ tends to $1$.

This differs significantly from the symmetric-group case, where almost every character-table entry is divisible by any fixed prime.

We also prove an angular equidistribution result for the nonzero character values as $q\to\infty$.

We show that the arguments become equidistributed in $[0,2\pi]$.

This proves an analogue of Miller's question on the distribution of signs among the nonzero entries in character tables of symmetric groups.