The $abc$ Conjecture Revisited

We propose a new abc-type conjecture. We motivate the conjecture and illustrate its relevance through several applications. Our main result concerns the function $$W(x,y) := \sum_{j = 1}^{y}\omega(x+j) \quad (y \in \mathbb{N},\ x \in \mathbb{Z}_{\ge 0})$$ where $\omega(n)$ denotes the number of distinct prime divisors of $n$. The new conjecture implies that, for each fixed $y \in \mathbb{N}$, $$\limsup_{x \to \infty} \frac{W(x,y)\log\log x}{\log x} = 1.$$