manuscripta mathematica

, Volume 151, Issue 3–4, pp 329–352 | Cite as

An improved analytic method for calculating \(\pi (x)\)

  • Jan BütheEmail author


We provide an improved version of the analytic method of Franke et al. for calculating the prime-counting function \(\pi (x)\), which is more flexible and, for calculations not assuming the Riemann Hypothesis, also more efficient than the original method. The new method has recently been used to calculate the value \(\pi (10^{25})=176,846,309,399,143,769,411,680\).

Mathematics Subject Classification

Primary 11Y35 Secondary 11Y70 


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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Hausdorff Center for MathematicsBonnGermany

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