I'm calculating the roots of the function \begin{equation} R(x) = \sum_{k=1}^{\infty}\frac{\mu(k)}{k}li(x^{1/k}) \end{equation} This function seems to have a largest and smallest positive root. Can anyone tell me if the roots of $R(x)$ have any significance for the prime counting function?

Not that I know of. If you have not already, see the paper by Folkmar Bornemann that describes a method for finding the roots of R(x) (see link below). It's a very interesting method. Best regards, Tom 

