Any ways to Simplify Daboussi's Argument for Prime Number Theorem?

One strategy to prove the Prime number theorem involves removing some factors:

$$\limsup_{x \to \infty} \underbrace{\frac{1}{x}\sum_{n \leq x} \mu(x)}_{\color{red}{A}}\leq \prod_{p \leq y} \left( 1 - \frac{1}{p}\right) \int_1^\infty \underbrace{ \frac{1}{t}\sum_{n \leq t} \mu(n)\mathbf{1}_y(n)}_{\color{blue}{B}} \times \frac{dt}{t}$$

The averages $A$ and $B$ are related since we are adding up the Möbius function on the subset of the integers whose factors $p \leq y$.

Here $\mathbf{1}_y(n) = 1$ if all the prime factors $p | n$ are less than $y$ and $0$ otherwise. Can't think of a better way of explaining this at the moment, it is called $v_y(n)$ in Daboussi's paper.

Proving that $\int_1^\infty \dots \ll \log y$ is rather convoluted. Are there any simpler ways to establish Daboussi's intermediate result?

• I don't know. Have you seen the exposition of Daboussi's proof in Tenenbaum and Mendes France, The Prime Numbers and Their Distribution, published by the AMS as Volume 6 of the Student Mathematical Library? – Gerry Myerson Oct 1 '15 at 1:10
• I think that while it is true that Daboussi’s proof is a nice application of what someone called ‘phipsiology’ (counting numbers without small/large prime factors), there is no way around the observation that Selberg’s elementary proof is simpler and easier to understand. – M Mueger Apr 21 at 8:46