An improved analytic Method for calculating $\pi(x)$
  We present an improved version of the analytic method for calculating $\pi(x)$, the number of prime numbers not exceeding $x$. We implemented this method in cooperation with J. Franke, T. Kleinjung and A. Jost and calculated the value $\pi(10^{25})$. 
Jan B\"uthe
math.NT math.NA