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24684Re: Twin primes and Pi

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  • Alexander
    Nov 26, 2012
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      --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
      >
      >
      >
      > --- In primenumbers@yahoogroups.com,
      > "djbroadhurst" <d.broadhurst@> wrote:
      >
      > > > Let t be the smaller partner of a prime twin (t, t+2)
      > > > calculate the infinite product P t/(t-1)
      > >
      > > Given the best current estimate of Brun's constant,
      > > I estimate your product as 3.19617816863...
      > > which is significantly greater than Pi.
      >
      > Off list, I was asked for my method.
      >
      > I took a log and subtracted half of Brun's constant, B.
      > Then one may truncate at modest values of t and use the
      > Hardy-Littlewood heuristic to approximate the remainder
      > as an incomplete gamma function:
      >
      > {S(t)=log(t/(t-1))-(t+1)/(t*(t+2));}
      >
      > {R(t)=3*0.66016181584686957*incgam(-1,log(t));}
      >
      > {B=1.902160583104;T=134217779;default(primelimit,T);
      > default(realprecision,13);e=2^21;s=0;
      > forprime(t=3,T,if(isprime(t+2),s+=S(t);
      > if(t>e,print([exp(s+B/2+R(t)),t]);e*=2)));}
      >
      > [3.196178168631, 2097257]
      > [3.196178168641, 4194581]
      > [3.196178168628, 8388617]
      > [3.196178168631, 16777289]
      > [3.196178168630, 33554501]
      > [3.196178168629, 67109321]
      > [3.196178168630, 134217779]
      >
      > There is no point in using more twin primes, since
      > the value of B is uncertain to more that 1 part in 10^13:
      > https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;9229e3b7.0208
      >
      > David
      >

      Impressing. But my very simple program calculates e.g.for
      t = 2097257 product = 2.9190232000697852233 ... and for
      t = 4194581 product = 2.931134127297162059 ...
      which is very different from your results.
      I am currently at
      t = 9118078559 product = 3.0173671938991475014 ...
      The program is tested several times. I would appreciate if you could also use a simple program without Brun's constant.

      Werner
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