This afternoon I tried to read and understand some sections of the paper *Some Applications of Diophantine Approximation* by R. Tijdeman procedding from Number Theory for the Millenium III, A K Peters (2002).

I wondered about the transcendence of infinite sums that involve the Möbius function, denoted in this post as $\mu(n)$ for integers $n\geq 1$ (see if you want the corresponding Wikipedia *Möbius function* or the MathWorld encyclopedia that also is very good). My background on linear forms in logarithms isn't good, and neither I have a good intuition about these theories. I know that certain series involving the Möbius function have a good and deep mathematical content.

Question.My belief is that for each polynomial $Q(x)$ with integer coefficients, that is $Q(x)\in\mathbb{Z}[X]$, such that for all $x>1$ is positive $Q(x)>0$ with degree $\deg(Q)\geq 2$ one has that the sum of the series $$\sum_{n=1}^{\infty}\frac{\mu(n)}{Q(n)}$$ is transcendent. Can you find a counterexample or provide some reasoning about the veracity of the statement?Many thanks.

If it is very difficult to find the counterexample, then please add what work can be done to know what about the veracity of my belief. I hope that my question is interesting, and that you can provide some answer.