Thread: Minimum (Number Theory Game) View Single Post
 2021-10-19, 14:47 #21 arbooker     "Andrew Booker" Mar 2013 22·23 Posts Interesting game. Maybe you know this already, but there is a simple proof that the Nullwertzahlen have density 0: any Nullwertzahl $$n$$ satisfies $$n\le2^{2^{\Omega(n)-1}}$$, so that $$\Omega(n)>\log\log(n)/\log(2)$$. As you pointed out in another post, $$\Omega(n)$$ is almost always close to $$\log\log{n}$$, so very few $$n$$ have so many prime factors. Following the proof of the Hardy-Ramanujan theorem, I think you can push this as far as a proof that $$W_0(x)=O(x/(\log{x})^\delta)$$ for some $$\delta>0$$, but I don't see how to get $$\delta=1$$ from just this.