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?



Do 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


[Paper](http://www-m3.ma.tum.de/m3old/bornemann/challengebook/AppendixD/waldvogel_problem_solution.pdf)


