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

TL;DR

This paper studies the distribution of nonzero residues in Pascal's triangle modulo a prime p, generalizing to Dirichlet characters, and provides explicit bounds on the error term in the distribution of residue occurrences.

Contribution

It extends known results on residue distribution in Pascal's triangle to Dirichlet characters and offers explicit error bounds for the asymptotic distribution.

Findings

Distribution of residues approaches uniformity as rows increase

Explicit bounds on the error term in residue distribution

Generalization to sequences defined by Dirichlet characters

Abstract

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 $A_p(r)$.