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arXiv · 2205.00972

Multiplicative functions resembling the M\"{o}bius funciton

Qingyang Liu
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Abstract

A multiplicative function $f$ is said to be resembling the M\"{o}bius function if $f$ is supported on the square-free integers, and $f(p)=\pm 1$ for each prime $p$. We prove $O$- and $\Omega$-results for the summatory function $\sum_{n\leq x} f(n)$ for a class of these $f$ studied by Aymone, and the point is that these $O$-results demonstrate cancellations better than the square-root saving. It is proved in particular that the summatory function is $O(x^{1/3+\varepsilon})$ under the Riemann Hypothesis. On the other hand it is proved to be $\Omega(x^{1/4})$ unconditionally. It is interesting to compare these with the corresponding results for the M\"{o}bius function.

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Qingyang Liu. 2022-04-26. Multiplicative functions resembling the M\"{o}bius funciton. https://arxiv.org/abs/2205.00972

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