On a M\"obius double sum

TL;DR

This paper analyzes a double sum involving the Möbius function and least common multiples, providing uniform bounds relevant to analytic number theory and zero-density estimates.

Contribution

It establishes uniform upper bounds for the sum across different ranges of X, especially near epsilon close to zero.

Findings

Derived bounds for the sum S_epsilon(X) for various X ranges.

Showed convergence of the sum even at epsilon=0.

Enhanced understanding of sums related to the Möbius function in number theory.

Abstract

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 M\"obius 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 $0^+$.