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EDIT: Corrected my previous answer, which contained an incorrect reduction of the problem to a result in the literature.

The answer is likely negative. By [1, Theorem 1.1], the partial sums of any multiplicative function with values $\pm 1$ supported on squarefree numbers are unbounded. As indicated by Will Sawin in the comments, it may be possible to adapt the proof to non-pretentious multiplicative functions with $|f(p)|=1$ ($p$ prime) supported on squarefrees, which applies to $\sqrt\mu$ in particular. However I would be cautious about this without carefully checking the details. The corresponding result for completely multiplicative functions (not supported on squarefrees) is known by Theorem 1.4 of [https://arxiv.org/pdf/1911.06265].

1 Marco Aymone, The Erdős discrepancy problem over the squarefree and cubefree integers, Mathematika 68 (2022), no. 1, pp. 51–73, doi 10.1112/mtk.12117.

EDIT: Corrected my previous answer, which contained an incorrect reduction of the problem to a result in the literature.

The answer is likely negative. By [1, Theorem 1.1], the partial sums of any multiplicative function with values $\pm 1$ supported on squarefree numbers are unbounded. As indicated by Will Sawin in the comments, it may be possible to adapt the proof to non-pretentious multiplicative functions with $|f(p)|=1$ ($p$ prime) supported on squarefrees, which applies to $\sqrt\mu$ in particular. However I would be cautious about this without carefully checking the details. The corresponding result for completely multiplicative functions (not supported on squarefrees) is known by Theorem 1.4 of [https://arxiv.org/pdf/1911.06265], so in particular if we replace $\sqrt\mu$ with $\sqrt\lambda$ (Liouville function) the answer to the question is negative.

1 Marco Aymone, The Erdős discrepancy problem over the squarefree and cubefree integers, Mathematika 68 (2022), no. 1, pp. 51–73, doi 10.1112/mtk.12117.

EDIT: Corrected my previous answer, which contained an incorrect reduction of the problem to a result in the literature.

The answer is likely negative. By [1, Theorem 1.1], the partial sums of any multiplicative function with values $\pm 1$ supported on squarefree numbers are unbounded. As indicated by Will Sawin in the comments, it may be possible to adapt the proof to non-pretentious multiplicative functions with $|f(p)|=1$ ($p$ prime), which applies to $\sqrt\mu$ in particular. However I would be cautious about this without carefully checking the details. The corresponding result for completely multiplicative functions (not supported on squarefrees) is known by Theorem 1.4 of [https://arxiv.org/pdf/1911.06265].

1 Marco Aymone, The Erdős discrepancy problem over the squarefree and cubefree integers, Mathematika 68 (2022), no. 1, pp. 51–73, doi 10.1112/mtk.12117.

EDIT: Corrected my previous answer, which contained an incorrect reduction of the problem to a result in the literature.

The answer is likely negative. By [1, Theorem 1.1], the partial sums of any multiplicative function with values $\pm 1$ supported on squarefree numbers are unbounded. As indicated by Will Sawin in the comments, it may be possible to adapt the proof to non-pretentious multiplicative functions with $|f(p)|=1$ ($p$ prime) supported on squarefrees, which applies to $\sqrt\mu$ in particular. However I would be cautious about this without carefully checking the details. The corresponding result for completely multiplicative functions (not supported on squarefrees) is known by Theorem 1.4 of [https://arxiv.org/pdf/1911.06265].

1 Marco Aymone, The Erdős discrepancy problem over the squarefree and cubefree integers, Mathematika 68 (2022), no. 1, pp. 51–73, doi 10.1112/mtk.12117.

EDIT: Corrected my previous answer, which contained an incorrect reduction of the problem to a result in the literature.

The answer is likely negative. By [1, Theorem 1.1], the partial sums of any multiplicative function with values $\pm 1$ supported on squarefree numbers are unbounded. As indicated by Will Sawin in the comments, it may be possible to adapt the proof to non-pretentious multiplicative functions with $|f(p)|=1$ ($p$ prime), which applies to $\sqrt\mu$ in particular. However I would be cautious about this without carefully checking the details. The corresponding result for completely multiplicative functions is known [enter link description here][2].

1 Marco Aymone, The Erdős discrepancy problem over the squarefree and cubefree integers, Mathematika 68 (2022), no. 1, pp. 51–73, doi 10.1112/mtk.12117.

EDIT: Corrected my previous answer, which contained an incorrect reduction of the problem to a result in the literature.

The answer is likely negative. By [1, Theorem 1.1], the partial sums of any multiplicative function with values $\pm 1$ supported on squarefree numbers are unbounded. As indicated by Will Sawin in the comments, it may be possible to adapt the proof to non-pretentious multiplicative functions with $|f(p)|=1$ ($p$ prime), which applies to $\sqrt\mu$ in particular. However I would be cautious about this without carefully checking the details. The corresponding result for completely multiplicative functions (not supported on squarefrees) is known by Theorem 1.4 of [https://arxiv.org/pdf/1911.06265].

1 Marco Aymone, The Erdős discrepancy problem over the squarefree and cubefree integers, Mathematika 68 (2022), no. 1, pp. 51–73, doi 10.1112/mtk.12117.