On a M\"obius double sum
Olivier Ramar\'e, Sebastian Zuniga Alterman

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 , which converges even in the case , where denotes the M\"obius function and is the least common multiple of and . 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 -functions. We establish uniform upper bounds for across various ranges of , with particular emphasis on the case close to .
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