= Positive density of power-smooth integers
{title2=$\#\{n\leq H:P^+(n)\leq H^\eta\}\gg_\eta H$}
Fix $\eta=1/k$ with $k\geq2$. Choose $k$ primes in $[H^{\eta-\eta^2},H^\eta]$, repetitions allowed, and an integer cofactor $m$ so that $m p_1\cdots p_k\leq H$. The product is between $H^{1-\eta}$ and $H$, so $m\leq H^\eta$ and all resulting integers are $H^\eta$-smooth. The <Mertens theorem for reciprocal primes> makes the reciprocal prime sum tend to $-\log(1-\eta)>0$. Since $\lfloor H/(p_1\cdots p_k)\rfloor\geq H/(2p_1\cdots p_k)$, there are $\gg_\eta H$ ordered representations. Each integer has only $O_\eta(1)$ prime factors in that interval, counted with multiplicity, so representation multiplicities are bounded by a constant depending on $k$. This proves the density bound. For arbitrary fixed positive $\eta$, choose a smaller reciprocal-integer exponent.
// Target: number-theory.bigb
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