 Euler product for a smoothed divisor-square correlation

For $\mathrm{Re}z,\mathrm{Re}{z}^{\prime }>0$, the series ${\sum }_{d,e}\mu (d)\mu (e)d^{-z}{e}^{-z^{\prime }}/[d,e]$ has local factor $1-p^{-1-z}-p^{-1-z^{\prime }}+p^{-1-z-z^{\prime }}$. Dividing by the displayed zeta function ratio leaves a holomorphic function $H$ near zero with $H(0,0)=1$. The local factor of $H$ is $(1-a-b+c)(1-c)/((1-a)(1-b))$, where $a=p^{-1-z}$, $b=p^{-1-z^{\prime }}$, $c=p^{-1-z-z^{\prime }}$, and equals one exactly at zero. With $z=(1+it)/\log X$, the zeta function poles produce the kernel $(1+it)(1+i{t}^{\prime })/((2+i(t+t^{\prime }))\log X)$ that determines the smooth divisor-square sieve asymptotic.