Proof in the literature of an equality involving the prime counting function - MathOverflow

Proof in the literature of an equality involving the prime counting function

2010-09-13T14:17:51Z

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.

Answer by Micah Milinovich for Proof in the literature of an equality involving the prime counting function

2010-09-13T16:04:08Z

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

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

Answer by Gerry Myerson for Proof in the literature of an equality involving the prime counting function

2010-09-14T04:27:23Z

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?