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?
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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
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$\begingroup$ The first lemma of this paper seems to making the assumption that the zeros of the zeta-function are all simple (without explicitly stating it). $\endgroup$ Commented Sep 16, 2010 at 23:08