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.
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It is quite obvious that $\zeta(3/2)$ and $\sum_{n=1}^{\infty}\frac{|\mu(n)|}{n^{\frac{3}{2}}} = \frac{\zeta(3/2)}{\zeta(3)}$ can't be better understood than $\zeta(3) $ (which appears in many expressions for example the 3rd derivative of $\log \Gamma(s)$ at $s=1$) – reuns Jul 08 '19 at 06:15