I was working with the irrationality of zeta values. I noticed that $$ \zeta(\frac{3}{2})=\zeta(3)\sum_{n=1}^{\infty}\frac{|\mu(n)|}{n^{\frac{3}{2}}} $$ So, as we know that $\zeta(3)$ is irrational (due to Apery), it is natural to study the sum $\sum_{n=1}^{\infty}\frac{|\mu(n)|}{n^{\frac{3}{2}}}$. I would like to know that this infinite sum is rational or irrational. Any progress will be highly appreciated.
2026-03-25 11:16:09.1774437369
The inverse infinite sum of square free numbers
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