We denote for integers $m\geq 1$ the Möbius function as $\mu(m)$. With the help of a CAS, Wolfram Alpha online calculator, I was calculating certain values of $$\sum_{n=0}^\infty\frac{(-1)^n\mu(2n+1)}{(2n+1)^k}\tag{1}$$ at integer arguments, for example $k=2,3$ and $k=4$. It is from the Möbius inversion of the Taylor series for the arctangent function and calculating integrals $$\int_0^1\frac{\arctan(z^{n+1})}{z}(\log z)^{k-2}dz$$ for the mentioned integers $k$.

**Example.** For example, it seems that $\sum_{n=0}^\infty\frac{(-1)^n\mu(2n+1)}{(2n+1)^4}=\frac{1536}{\psi^{(3)}(\frac{1}{4})-\psi^{(3)}(\frac{3}{4})}$, where $\psi^{(3)}(z)$ is the polygamma function of order $3$ (it is the notation from the Wikipedia *Polygamma function*).

Question 1.I am curious to know if these particular values are in the literature and how to calculate them. Please if it is in the literature refers it and I try to search and read it from the literature, in other case can you express in terms of particular values of well-known functions and constants for example the case $k=5$ or $k=6$ of $(1)$?Many thanks.

Let $s=x+iy$ the complex variable, and we consider $$\eta(s):=\sum_{n=0}^\infty\frac{(-1)^n\mu(2n+1)}{(2n+1)^s}.$$

Then it is easy to see that $\eta(s)$ converges absolutely for $\Re s >1$, since by comparison with the harmonic series, $|\sum_{n=0}^\infty\frac{(-1)^n\mu(2n+1)}{(2n+1)^s}|\leq \sum_{n=0}^\infty\frac{(-1)^n\mu(2n+1)}{(2n+1)^{\Re s}}.$

Question 2.Can you provide useful hints or ideas to know what about the abscissa of convegence of our series $\eta(s)$? Please if it is in the literature refers it and I try to search and read it from the literature.Many thanks.

By the methods of analytic number theory for Dirichlet series, I think that should be useful calculate the asympotitc behaviour of $$\sum_{1\leq n\leq x}(-1)^{n}\mu(2n+1),$$ but I don't know nothing about it.

**Footnote (See comments).** Greg Martin anticipated a follow up question (
I thought to ask about the possibility of a functional equation and what about the zeros).