I'm working on Ritcher's "A new elementary proof of the Prime Number Theorem" paper. I have some doubt about the proof of proposition 3.1 Here's the reference to the paper: https://arxiv.org/abs/2002.03255

What I found not clear is the bound at the beginning of p.7:

Lemma 3.3 (applied with $\sigma = 2$) gives the estimate: $$\bigg|P \cap\left(\frac{8^x}{2^{n+1}}, \frac{8^x}{2^n}\right]\bigg| \leq \frac{8^x}{2^{n+1}x} + O(1).$$

It is clearly not the direct application of Lemma 3.3 and I'm not sure how to get this bound starting from the one that Lemma 3.3 actually gives.

Bonus: it's less relevant to me at the moment but I'm also struggling to understand how the term $O(1/N)$ arises in the first equality in “Proof of Theorem 1.1 assuming Proposition 2.2.” at p.4