The real number given by the absolutely convergent series


$$\displaystyle A = \sum\_{k=1}^\infty \frac{|\mu(k)|}{k \phi(k)}$$


is known as Landau's Totient Constant. It can be explicitly evaluated to be $\frac{\zeta(2)\zeta(3)}{\zeta(6)}$. Indeed, we see that $A$ can be expanded into an Euler product


$$\displaystyle A = \prod\_p \left(1 + \frac{1}{p(p-1)} \right).$$


We then have


\begin{align\*} 1 + \frac{1}{p(p-1)} & = \frac{p^2 - p + 1}{p(p-1)} \\
& = \frac{p^3 + 1}{p(p^2 - 1)} \\
& = \frac{(p^3 - 1)(p^3 + 1)}{p(p^2 - 1)(p^3 - 1)} \\
& = \frac{p^6 - 1}{p(p^2 - 1)(p^3 - 1)} \\
& = \frac{1 - p^{-6}}{(1 - p^{-2})(1 - p^{-3})},
\end{align\*}
and from here we see that


$$\displaystyle \prod\_p \left(1 + \frac{1}{p(p-1)} \right) = \prod\_p \left(\frac{1 - p^{-6}}{(1 - p^{-2})(1 - p^{-3})} \right) = \frac{\zeta(2) \zeta(3)}{\zeta(6)}$$


as desired.


I am looking to evaluate the related number


$$\displaystyle B = \sum\_{k=1}^\infty \frac{\mu(k)}{k \phi(k)} = \prod\_p \left(1 - \frac{1}{p(p-1)} \right),$$


where there is no absolute value in the numerator. The "complete the cube" trick used above obviously will not work, as one gets the quadratic polynomial $x^2 - x - 1$ instead in the numerator. Is there a nice expression for $B$? In lieu of that, is there a reasonable interpretation for $B$?



A number similar to Landau's Totient constant
This constant is known as [Artin's Constant.](https://mathworld.wolfram.com/ArtinsConstant.html) The book *Finch S. R. Mathematical constants* (section 2.4) gives the following information.



> 
> A rapidly convergent expression for Artin's constant is as follows [12-18]. Define Lucas' sequence as $$
> l\_{0}=2, \quad l\_{1}=1, \quad l\_{n}=l\_{n-1}+l\_{n-2} \quad \text { for } n \geq 2
> $$
> and observe that $l\_{n}=\varphi^{n}+(1-\varphi)^{n},$ where $\varphi$ is the Golden mean [1.2]. Then
> $$
> \begin{aligned}
> C\_{\text {Artin }} &=\prod\_{n \geq 2} \zeta(n)^{-\frac{1}{n} \sum\_{k \mid n} l\_{k} \cdot \mu\left(\frac{n}{k}\right)} \\
> &=\zeta(2)^{-1} \zeta(3)^{-1} \zeta(4)^{-1} \zeta(5)^{-2} \zeta(6)^{-2} \zeta(7)^{-4} \zeta(8)^{-5} \zeta(9)^{-8} \cdots.
> \end{aligned}
> $$
> 
> 
> 


