 Smooth divisor-square sieve asymptotic

For a real smooth cutoff $\varphi$ supported on $[-1/3,1/3]$ with $\varphi (0)=1$, define $F_X(n)=({\sum }_{d\mid n}\mu (d)\varphi (\log d/\log X){)}^2$. Its sum over any interval of length $X$ is $c_{\varphi }X/\log X+O_{\varphi }(X/{\log }^{2}X+X^{2/3})$, uniformly in the interval location. Here $c_{\varphi }$ 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.