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Statement

Let $N$ be sufficiently large and $3\le y<z\le\log N$. If $X$ is the set of positive integers divisible by no prime in $[y,z]$, then

$$|X\cap[N,2N)|\ll N\frac{\log y}{\log z}.$$

Source. Bloom, arXiv:2112.03726v2, Lemma 1, printed/PDF p. 6.

Rewritten proof

Let $P$ be the product of the primes in $[y,z]$. Inclusion-exclusion over its squarefree divisors gives

$$\begin{aligned} |X\cap[N,2N)| &=\sum_{d\mid P}\mu(d)\left(\frac Nd+O(1)\right)\\ &=N\prod_{y\le p\le z}\left(1-\frac1p\right)+O(2^{\pi(z)}). \end{aligned}$$

Endpoint rounding changes each count by at most an absolute constant. The Mertens product estimate gives a main term $\ll N\log y/\log z$. Also $2^{\pi(z)}\le2^z\le N^{\log2}$, which is $o(N/\log\log N)$; since $\log y/\log z\gg1/\log\log N$, the error is absorbed. This proves the bound.

The same inclusion-exclusion calculation on $[1,T]$ gives the bound $\ll T\log y/\log z$ whenever $z\le\log T$. This variant is used in Lemma 2.

Dependencies

The external Mertens estimate $\prod_{p\le x}(1-1/p)^{-1}\asymp\log x$ is equation (2) on p. 3; Bloom cites Montgomery and Vaughan, Multiplicative Number Theory I, Chapter 2. Bloom also cites their Theorem 3.1 for inclusion-exclusion.

Bears on

• Problem 298
• Problem 299