Asymptotic Distribution of Residues in Pascal's Triangle mod p

2023/09/22 by Lane, Connor
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2309.12942

Abstract

Fix a prime p and define T_p(n) to be the number of nonzero residues in the nth row of pascal's triangle mod p, and define ϕ_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_χ(n) and ϕ_χ(n) for a Dirichlet character χ of modulus p. We prove many properties of these sequences that generalize those of T_p(n) and ϕ_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)∼ (1)/(p-1)ϕ_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).

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