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Re: Conjecture Ludovicus V [HL puzzle]

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  • djbroadhurst
    ... Here are more than 500 good digits of the relevant Hardy-Littlewood contant 0.686406731409123004556096348363509434089166550627
    Message 1 of 6 , Nov 12, 2012
      --- In primenumbers@yahoogroups.com,
      "djbroadhurst" <d.broadhurst@...> wrote:

      > 0.68640673...

      Here are more than 500 good digits of the relevant
      Hardy-Littlewood contant

      0.686406731409123004556096348363509434089166550627\
      87977896811707366392111335868511586385990346954399\
      18910971210114370836967066905693197374076733762228\
      99676181590249072148452433271956678446776133507821\
      66911238709121761546057993037867560421525052339604\
      30362058008970065288339284308924109839850906244857\
      05610181784829261874808914954658125072564596078857\
      97178488488284913596339147096072562959833915384281\
      95797768815771346844648575433020873782746429273861\
      06852378388023207827786598975144050262059586826459\
      0665546

      obtained in about 3 GHz-minutes.

      Puzzle: What are the next 6 decimal digits?

      David
    • mikeoakes2
      ... Regarding that nice 1960 paper: Daniel Shanks describes his eratosthenese-type sieve for factoring numbers of the form n^2+1 here:
      Message 2 of 6 , Nov 12, 2012
        --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
        >
        > --- In primenumbers@yahoogroups.com,
        > Robert Gerbicz <robert.gerbicz@> wrote:
        >
        > > 0.5*prodprime(p=3,inf,1+(-1)^((p+1)/2)/(p-1))
        >
        > Nice work, Robert.
        >
        > http://tech.groups.yahoo.com/group/primenumbers/message/24612
        > http://www.ams.org/journals/mcom/1960-14-072/S0025-5718-1960-0120203-6/S0025-5718-1960-0120203-6.pdf
        >
        > 0.68640673...

        Regarding that nice 1960 paper: Daniel Shanks describes his eratosthenese-type sieve for factoring numbers of the form n^2+1 here:
        http://www.ams.org/journals/mcom/1959-13-066/S0025-5718-1959-0105784-2/S0025-5718-1959-0105784-2.pdf

        Mike
      • djbroadhurst
        ... Here, Shanks remarks that ... To get more than 500 decimal places of the HL constant, in less than 80 GHz-seconds, it suffices to consider primes up to
        Message 3 of 6 , Nov 12, 2012
          --- In primenumbers@yahoogroups.com,
          "mikeoakes2" <mikeoakes2@...> wrote:

          > http://www.ams.org/journals/mcom/1959-13-066/S0025-5718-1959-0105784-2/S0025-5718-1959-0105784-2.pdf

          Here, Shanks remarks that

          > With this improvement, the first two p's,
          > i.e., 5 and 13, already suffice to yield
          > the five decimal places shown.

          To get more than 500 decimal places of the HL constant, in
          less than 80 GHz-seconds, it suffices to consider primes up
          to 31601 and to augment that data with the values of
          beta(2*k) = suminf(n=0,(-1)^n/(2*n+1)^(2*k))
          for k = 1 to 60. Note however that Pari-GP's "zetak" is
          broken at this high precision, so one needs a few lines of
          code, calling "incgam", for the summand of Proposition 3.2
          of "High precision computation of Hardy-Littlewood
          constants" by Henri Cohen, available as a .dvi file from
          http://www.math.u-bordeaux1.fr/~hecohen/

          David
        • djbroadhurst
          ... Solution: 141322 Method: Using incgam , eint1 and moebius this solution may be found in less than 8 seconds, running at less than 3 GHz:
          Message 4 of 6 , Nov 17, 2012
            --- In primenumbers@yahoogroups.com,
            "djbroadhurst" <d.broadhurst@...> wrote:

            > Here are more than 500 good digits of the relevant
            > Hardy-Littlewood constant
            >
            > 0.686406731409123004556096348363509434089166550627\
            > 87977896811707366392111335868511586385990346954399\
            > 18910971210114370836967066905693197374076733762228\
            > 99676181590249072148452433271956678446776133507821\
            > 66911238709121761546057993037867560421525052339604\
            > 30362058008970065288339284308924109839850906244857\
            > 05610181784829261874808914954658125072564596078857\
            > 97178488488284913596339147096072562959833915384281\
            > 95797768815771346844648575433020873782746429273861\
            > 06852378388023207827786598975144050262059586826459\
            > 0665546
            >
            > Puzzle: What are the next 6 decimal digits?

            Solution: 141322

            Method: Using "incgam", "eint1" and "moebius" this solution may
            be found in less than 8 seconds, running at less than 3 GHz:

            {F(n,k)=local(z=C*n^2);
            incgam(1/2+k,z)/n^(2*k)+n*S[k]*eint(k,exp(-z),z);}

            {eint(k,e,x)=if(k==1,eint1(x),(e-x*eint(k-1,e,x))/(k-1));}

            {a(s)=sumdiv(s,d,if(d%2,moebius(d)*2^(s/d),0))/(2*s);}

            {b(s)=if(s==1,0,a(s)-if(s%2,0,a(s/2)));}

            {default(realprecision,550);P=31601;terms=60;gettime;
            got=505;ext=6;C=Pi/4;S=sqrt(C)*vector(terms,k,C^k);
            N=2*terms+1;Z=vector(N,s,if(s>1,zeta(s)*(1-1/2^s),1));
            V=Vec(Pi/(4*cos(Pi*x/2+O(x^(N+2)))));L=vector(N,s,if(s%2,V[s],
            suminf(n=0,F(4*n+1,s/2)-F(4*n+3,s/2))/gamma((s+1)/2)));t=0.5;
            forprime(p=3,P,z=(-1)^((p-1)/2);t*=1-z/(p-1);r=1.;for(s=1,N,r/=p;
            L[s]*=1-z*r;Z[s]*=1-(s>1)*r));for(s=1,N,t/=L[s]^a(s)*Z[s]^b(s));
            t=floor(10^(got+ext)*t);print(t%(10^ext)" in " gettime" ms");}

            141322 in 7507 ms

            David
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