We express $\mathrm{\Delta }\left(x,N\right)$, as defined in the title, for $x=\frac{a}{q}$ and $q$ prime in terms of values of characters modulo $q$. Using this, we show that the universal lower bound for $\mathrm{\Delta }\left(N\right)={sup}_{x\in \mathbb{R}}\left|\mathrm{\Delta }\left(x,N\right)\right|$ can, in general, be substantially improved when $N$ is composed of primes lying in a fixed residue class modulo $q$. We also prove a corresponding improvement when $N$ is the product of the first s primes for infinitely many natural numbers $s$.

In this paper we consider an extension to friable integers of the arcsine law for the mean distribution of the divisors of integers, originally due to Deshouillers, Dress and Tenenbaum.We describe the limit law and show that it departs from the arcsine law when the friability parameter $u:=logx/logy$ increases. More precisely, as $u\to \infty$, the mean distribution shifts from the arcsine law towards Gaussian behaviour.