Symmetric Dyck Paths and Hooley’s $\varDelta$-Function

Abstract

Hooley [6] introduced the function

$$ \varDelta (n) := \max _{u \in \mathbb {R}}\# \left\{ d | n: \quad u < \log d \leqslant u+1 \right\} , $$

where $\log$ is the natural logarithm. Changing the base of the logarithm from e to an arbitrary real number $\lambda > 1$, we define

$$ \varDelta _{\lambda }(n) := \max _{u \in \mathbb {R}}\# \left\{ d | n:\quad u < \log _{\lambda } d \leqslant u+1 \right\} . $$

The aim of this paper is to express $\varDelta _{\lambda }(n)$ as the height of a symmetric Dyck path defined in terms of the distribution of the divisors of n.