In their article "Calculations Related to Riemann's Prime Number Formula", Riesel and Gohl claim a simplification of Riemann's evaluation of the modified prime counting function \begin{align} \pi_{0}(x) & = \tfrac{1}{2} \lim_{\epsilon \to 0} \pi(x + \epsilon) + \pi(x - \epsilon) \\ & = \lim_{N \to \infty} R_{N}(x) - F_{N}(x) \end{align} where $F_{N}(x)$ is an infinite sum which depends only on the complex zeros of the zeta function in the critical strip and accounts for the discontinuities of $\pi_{0}$ at the primes, while \begin{align} R_{N}(x) = \sum_{n \geq 1} \frac{\mu(n)}{n} \mathsf{li}(\sqrt[n]{x}) + \frac{1}{2 \log x} \sum_{m = 1}^{N} \mu(n) + \frac{1}{\pi} \arctan \frac{\pi}{\log x} + \epsilon_{N}(x) \end{align} is a smooth function of $x$, and $\epsilon_{N} \to 0$ as $N \to \infty$. The first term is Riemann's well-known approximation to $\pi_{0}$, and the remaining terms depend only on the trivial zeros of the zeta function (at the negative even integers). One often finds the approximation, \begin{align} \pi_{0}(x) \approx \sum_{n \geq 1} \frac{\mu(n)}{n} \mathsf{li}(\sqrt[n]{x}) - \frac{1}{ \log x} + \frac{1}{\pi} \arctan \frac{\pi}{\log x} \end{align} usually attributed to them. What of the factor of $\frac{1}{2}$ and sum involving the Möbius function; doesn't this sum diverge?
Riesel and Gohl's Approximation of the Modified Prime Counting Function, $\pi_{0}$
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real-analysis
number-theory