On a Möbius double sum

Abstract

We study the double sum $S_{\epsilon }(X)$$=$${\sum }_{\begin{array}{c}d,e\le X\end{array}}\frac{\hat{I}¼(d)\hat{I}¼(e)}{[d,e{]}^{1+\epsilon }}$, which converges even in the case $\epsilon =0$, where $\hat{I}¼$ 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 $S_{\epsilon }(X)$ across various ranges of $X$, with particular emphasis on the case $\epsilon$ close to $0^{+}$.