Symmetric Dyck Paths and Hooley's Δ-Function

摘要

Hooley [6] introduced the function Δ(n) :=max # {d|n : u < log d ≤ u + 1}, u∈R where log is the natural logarithm. Changing the base of the logarithm from e to an arbitrary real number λ > 1, we define Δ_λ(n) := max # {d|n: u < log_λ d ≤ u + 1}. u∈R The aim of this paper is to express Δ_λ(n) as the height of a symmetric Dyck path defined in terms of the distribution of the divisors of n.