Let \begin{equation} R(x) = \sum_{k=1}^{\infty}\frac{\mu(k)}{k}li(x^{1/k}) \end{equation} where $\mu$ is the Mobius function and \begin{equation} li(x) = \int_0^x \frac{dt}{\log t} \end{equation} Is there a proof in the literature of \begin{equation} \pi(x)=R(x)-\sum_{\rho}R(x^{\rho}) \end{equation} where $\pi$ is prime counting function and the sum is over all complex zeros of $\zeta(s)$. The literature seems to treat it as fact while stating no proof is available - a strange situation.

Thanks in advance.

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    $\begingroup$ This link is on the Wiki article jstor.org/pss/2004630 $\endgroup$ – Gjergji Zaimi Sep 13 '10 at 14:45
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    $\begingroup$ I think Edwards' book <i>The Zeta Function</i> has a proof of this, but I don't have it available to check. $\endgroup$ – David E Speyer Sep 13 '10 at 14:54
  • $\begingroup$ I think there is an $O(1)$ missing. This is a formula stated by Riemann in "the paper" and proved by von Mangoldt. As David says, it's in chapter 3 of Edwards. $\endgroup$ – Felipe Voloch Sep 14 '10 at 18:30

Stopple, A Primer of Analytic Number Theory, proves a theorem which looks something like the one under discussion. On page 248, he has $$\pi(x)=R(x)+\sum_{\rho}R(x^{\rho})+\sum_{n=1}^{\infty}{\mu(n)\over n}\int_{x^{1/n}}^{\infty}{dt\over t(t^2-1)\log t}$$

You say that the literature treats your formula as a fact, but you give no citation. Where in the literature do you find your formula?

  • $\begingroup$ Borwein, Computational Strategies for the Riemann Zeta Function. This has the formula I quote and claims that it is exact. $\endgroup$ – alext87 Sep 14 '10 at 11:19
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    $\begingroup$ If Borwein is interested in computing $\pi (x)$ then he might be taking advantage of the fact that it is an integer and the error term that he is omitting is small. $\endgroup$ – Felipe Voloch Sep 14 '10 at 18:33
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    $\begingroup$ alext in the original question refers to a sum over 'all complex zeros'; it's not clear if he's referring to the trivial zeros or not. If he is, I think what he wrote is correct. The formula in my book that Gerry references has a sum over the nontrivial zeros only; the other infinite sum on n is the contribution of the trivial zeros, written in an explicit form to make clear that it is very small (decreasing in x). $\endgroup$ – Stopple Sep 14 '10 at 20:14
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    $\begingroup$ I've looked at the Borwein (and Bradley and Crandall) paper - a preprint is freely available at docserver.carma.newcastle.edu.au/211/2/… and where alext87 has equality Borwein et al. have a tilde. They say, "This relation has been called exact [by Ribenboim, in The New Book of Prime Number Records], yet we could not locate a proof in the literature." alext87, if you had access to the reference and the full quote, it would have saved us a lot of work had you included the information in your original question. $\endgroup$ – Gerry Myerson Sep 14 '10 at 23:29
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    $\begingroup$ @Gerry - I think the plus signs in my book are a typo; they should in fact be minus signs! The corresponding formula for $\Pi(x)$, (10.11) on p. 244, has minus. $\endgroup$ – Stopple Oct 4 '10 at 20:52

It may be useful to read Section 10 of Chapter V of Ingham's "The Distribution of Prime Numbers."

Let $\Pi(x)=\pi(x)+\frac{1}{2}\pi(x^{1/2})+...$, then Moebius proved that

$$ \pi(x) = \sum_{n=1}^\infty \frac{\mu(n)}{n} \Pi(x^{1/n}).$$

However, this isn't overly illuminating because it shows that

$$\pi(x) = \Pi(x) + O(\sqrt{x}/\log x ). $$

and Littlewood showed that

$$ \Pi(x) - \ell i(x) = \Omega_\pm(\sqrt{x} \log\log\log x/\log x).$$


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