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Source. Definition and Lemma 3, printed pp. 144–145 (PDF pp. 2–3).

For a fixed positive integer $r$, define the coprime factors

$$h_r(n)=\prod_{\substack{p^\alpha\parallel n\\\alpha>r}}p^\alpha, \qquad l_r(n)=\prod_{\substack{p^\alpha\parallel n\\\alpha\le r}}p^\alpha.$$

Thus $n=h_r(n)l_r(n)$, and either factor may be one. Write $\Omega(n)=\sum_{p^\alpha\parallel n}\alpha$, with $\Omega(1)=0$.

Statement. Fix $r\ge1$ and $c>0$. As $x\to\infty$, the number of $n\le x$ satisfying $n=l_r(n)$ and $\Omega(n)>c\sqrt{\log x/\log\log x}$ is at most

$$x\exp\left(-\left(\frac c2-o(1)\right) \sqrt{\log x\log\log x}\right).$$

The error may depend on fixed $r,c$.

Complete proof

Put $B=\sqrt{\log x/\log\log x}$, $T=\sqrt{\log x\log\log x}$ and $A=\sum_{p\le x}1/p$. The elementary prime-reciprocal bound gives $A=O(\log\log x)$; its proof is already supplied in the Croot unit.

Separate the counted integers into those with $\omega(n)>cB$ and the remaining set $\mathcal T$. The first class satisfies the claimed bound by Lemma 2. For $n=\prod p^{\alpha_p}$ set $g(n)=\prod\alpha_p!$. The multinomial expansion, keeping just products at most $x$, gives

$$\sum_{\substack{n\le x\\\Omega(n)=j}}\frac1{g(n)n} \le\frac{A^j}{j!}.$$

Let $J=\lfloor cB\rfloor+1$. For all large $x$, $J>2A$. Successive terms of $A^j/j!$ for $j\ge J$ have ratio at most $1/2$, so

$$\sum_{\substack{n\le x\\\Omega(n)>cB}}\frac1{g(n)n} \le \sum_{j\ge J}\frac{A^j}{j!} \le \frac{2A^J}{J!}.$$

The integral estimate $\log J!\ge J\log J-J$ now yields

$$\log\frac{2A^J}{J!} \le -J\log J+J\log A+J+O(1) =-\left(\frac c2+o(1)\right)T.$$

Indeed $\log J=\tfrac12\log\log x+O(\log\log\log x)$, $J=(c+o(1))B$, and $\log A=O(\log\log\log x)$.

For $n\in\mathcal T$, all $\alpha_p\le r$ and $\omega(n)\le cB$. Consequently

$$g(n)\le(r!)^{\omega(n)}\le(r!)^{cB}.$$

Since $n\le x$,

$$|\mathcal T| \le x\sum_{n\in\mathcal T}\frac1n \le x(r!)^{cB}\sum_{n\in\mathcal T}\frac1{g(n)n} \le x\exp\left(-\left(\frac c2-o(1)\right)T\right).$$

The extra logarithm $cB\log(r!)$ is $o(T)$ because $r$ is fixed. Adding the first class absorbs a factor two into the $o(1)$ term.

This supplies the exponential-series tail estimate left abbreviated on p. 145. It includes $r=1$ and the strict-threshold integer endpoints. It makes no claim with $r$ growing with $x$.