On a M\"obius double sum

We study the double sum $S_\varepsilon(X)$$=$$\sum_{\substack{d,e\le
X}}\frac{\mu(d)\mu(e)}{[d,e]^{1+\varepsilon}}$, which converges even in the case $\varepsilon=0$, where $\mu$ denotes the Möbius function and $[d,e]$ is the least common multiple of $d$ and $e$.
Such expressions arise naturally in analytic number theory, notably as the diagonal contribution in certain squared mean values, and they play a significant role in zero-density estimates for the Riemann zeta function and related $L$-functions. We establish uniform upper bounds for $S_\varepsilon(X)$ across various ranges of $X$, with particular emphasis on the case $\varepsilon$ close to $0^+$.