Let for any two reals $ x $ and $ y $ such that $x\leq y $ the quantity $\mathcal{N}_{\alpha}(x,y) $ defined as $ \sum_{x\leq n\leq y}\Omega(n)^{\alpha} $ where $\Omega(n) $ is the total number of prime factors of $ n $ counted with multiplicity. One has $\lim_{\alpha\to -\infty} \mathcal{N}_{\alpha}(x,y)=\pi(y)-\pi(x) $. Moreover for given $ x $ and $ y $ the map $ \alpha\mapsto\mathcal{N}_{\alpha}(x,y) $ is increasing. I would thus be interested in an asymptotics for both $ \mathcal{N}_{\alpha}(x,y) $ and it derivative w.r.t $ \alpha $ in terms of $ \alpha $ , $ x $ and $ y $.
Has such a function been considered before ?
For $k\in \Bbb N$, let $\pi_k(x)$ be the number of integers $n\le x$ such that $\Omega(n)=k$. Observe that $\pi_1=\pi$. Then $$ \mathcal{N}_\alpha(2,y)=\sum_{k=1}^{\lfloor\log_2y\rfloor}2^\alpha\pi_k(y). $$ For fixed $y$, as $\alpha\to-\infty$, we have $$ \mathcal{N}_\alpha(2,y)=\pi(y)+2^\alpha\pi_2(y)+O(3^\alpha). $$ It is not difficult to see that $$ \pi_2(y)=\sum_{k=1}^{\pi(\sqrt n)}\pi(y/p_k)-\frac12\pi(\sqrt n)\bigl(\pi(\sqrt n)-1\bigr), $$ here $p_k$ is the $k$-th prime. Asymptotically $$ \pi_2(y)\sim y\,\frac{\log\log y}{\log y}. $$
For the general case, consider $\mathcal{N}_\alpha(2,y)-\mathcal{N}_\alpha(2,x)$.