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What is the asymptotic order of numbers divisible by no primes except those of the form $an+b$ ($a$, $b$ fixed)?

Surely (except for the trivial cases) they are of order strictly between that of he primes and of all numbers.

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    Given a subset of the primes $\mathcal{S}$ with positive density $\alpha$, I believe the asymptotics for the counting function for the integers formed by products of those primes will be $\sim C \frac{x}{(\log x)^{1-\alpha}},$ for some constant $C$. The idea is that the function $\zeta_\mathcal{S}(s)=\prod_{p\in\mathcal{S}} (1-p^{-s})^{-1}$ will have a singularity of the form $\frac{1}{(s-1)^{\alpha}}$ and then integrating $ \frac{x^s}{(s-1)^{\alpha}}dx$ gives a main term of $\frac{x}{(\log x)^{1-\alpha}}.$2012-04-17

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