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Re: What if Riemann's prime-counting formula was not the best?

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  • djbroadhurst
    ... No. The Gram formula is still very convenient at this size. Pari-GP, gives the exact value of R(10^250) in 0.1 seconds:
    Message 1 of 16 , Jul 30, 2013
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      --- In primenumbers@yahoogroups.com,
      Chroma <chromatella@...> wrote:

      >> R(x)=round(1+suminf(k=1,log(x)^k/(zeta(k+1)*k*k!)));
      > For large values of x, this algorithm is inconvenient,
      > eg for x = 10^250 requires over 1868 terms

      No. The Gram formula is still very convenient at this size.
      Pari-GP, gives the exact value of R(10^250) in 0.1 seconds:

      R(x)=round(1+suminf(k=1,log(x)^k/(zeta(k+1)*k*k!)));

      {default(realprecision,260);print(R(10^250));
      print(" took "gettime" milliseconds");

      17402062546569168467749416650483864101780289759689292646552693950034847365084787720410883002915274182213664956284195372937010842285191263145767899389242017061947571038442681072462756632213511422607548574658029047365218974809766827365028215685475746
      took 98 milliseconds

      Perhaps you are paying for inferior software?
      If so, the general rule is: the less you pay,
      the better the deal.

      Pari-GP is totally free and hence rather hard to beat :-)

      David
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