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arXiv · 2603.25961

On a M\"obius double sum

Olivier Ramaré·Sebastian Zuniga Alterman
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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 $0^+$.

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Olivier Ramaré, Sebastian Zuniga Alterman. 2026-03-26. On a M\"obius double sum. https://arxiv.org/abs/2603.25961

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