Conditional Calculation of pi(1024) 
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Email from Jens Franke [Thu 7/29/2010 2:47 PM]: (color added)

Using an analytic method assuming (for the current calculation) the Riemann Hypthesis, we found that the number of primes below 10^24 is 18435599767349200867866. The analytic method used is similar to the one described by Lagarias and Odlyzko, but uses the Weil explicit formula instead of complex curve integrals. The actual value of the analytic approximation to pi(10^24) found was 18435599767349200867866+3.3823e-08.

For the current calculation, all zeros of the zeta function below 10^11 were calculated with an absolute precision of 64 bits.

We also verified the known values of pi(10^k) for k<24, also using the analytic method and assuming the Riemann hypothesis.

Other calculations of pi(x) using the same method are (with the deviation of the analytic approximation from the closest integer included in
parenthesis)

pi(2^76)=1462626667154509638735 (-6.60903e-09)
pi(2^77)=2886507381056867953916 (-1.72698e-08)

Computations were carried out using resources at the Institute for Numerial Simulation and the Hausdorff Center at Bonn University. Among others, the programs used the GNU scientific library, the fftw3-library and mpfr and mpc, although many time critical floating point calculations were done using special purpose routines.

J. Buethe
J. Franke
A. Jost
T. Kleinjung

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