= Smooth divisor-square sieve asymptotic
{title2=$\sum_{n\in I}F_X(n)\sim c_\phi X/\log X$}
For a real <smooth> cutoff $\phi$ supported on $[-1/3,1/3]$ with $\phi(0)=1$, define $F_X(n)=(\sum_{d\mid n}\mu(d)\phi(\log d/\log X))^2$. Its sum over any interval of length $X$ is $c_\phi X/\log X+O_\phi(X/\log^2X+X^{2/3})$, uniformly in the interval location. Here $c_\phi$ is the <derivative energy constant for a smooth sieve cutoff>. Expanding the square yields an interval-counting error $O(X^{2/3})$ and an <Euler product for a smoothed divisor-square correlation>; its <zeta function> pole provides the main term. Every <prime> exceeding $X^{1/3}$ has weight one, giving the <short-interval prime upper bound from a smooth divisor weight>.
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