Asymptotic Distribution of Residues in Pascal's Triangle mod $p$

Fix a prime $p$ and define $T_p(n)$ to be the number of nonzero residues in the $n$th row of pascal's triangle mod $p$, and define $\phi_p(n)$ to be the number of nonzero residues in the first $n$ rows of pascal's triangle mod $p$. We generalize these to sequences $T_\chi(n)$ and $\phi_\chi(n)$ for a Dirichlet character $\chi$ of modulus $p$. We prove many properties of these sequences that generalize those of $T_p(n)$ and $\phi_p(n)$. Define $A_n(r)$ to be the number of occurrences of $r$ in the first $n$ rows of Pascal's triangle mod $p$. Guy Barat and Peter Grabner showed that for all primes $p$ and nonzero residues $r$, $A_n(r)\sim \frac{1}{p-1}\phi_p(n)$. We provide an alternative proof of this fact that yields explicit bounds on the error term. We also discuss the distribution of $A_p(r)$.