On the convergence of $\sum_{n=1}^\infty\log \left(\frac{\mu(n)}{\sqrt{n}}+1-\frac{1}{4n^2}\right)$, where $\mu(n)$ is the Möbius function

Question

Let $\mu(n)$ the Möbius function, see in MathWorld the definition if you need it.

I can not solve this example, and I would like to know if it's possible deduce an explanation about the convergence of the series (or as (the related) an infinite product with this criteria of Wikipedia, note that $a_1\geq 1>0$ and for $n\geq 2$ one has $\mu(n)/\sqrt{n}+1-\frac{1}{4n^2}\geq -1/\sqrt{n}+1-\frac{1}{4n^2}>0$).

Question. Please discuss if the following series is convergent $$\sum_{n=1}^\infty\log \left(\frac{\mu(n)}{\sqrt{n}}+1-\frac{1}{4n^2}\right).$$

Many thanks.

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Edit: I was wrong with my computational evidence and problems, I unerstand better the problem now, if there are some user that want solve the question is very welcome. Thanks.