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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?

edited Sep 13 2010 at 14:25

Charles
4,189●8●27

asked Sep 13 2010 at 14:24

alext87
1,508●3●14

Tom Dickens · Accepted Answer · 2010-09-16 22:52:49Z UTC

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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

Paper

answered Sep 16 2010 at 22:52

Tom Dickens
693●4●8

	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). – Micah Milinovich Sep 16 2010 at 23:08
	Okay. Thank you very much. – alext87 Sep 17 2010 at 7:09

Do the roots of R(x) have any significance for the prime counting function?

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