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Re: Sum of x/log(x) x=2..n

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  • David Broadhurst
    ... http://physics.open.ac.uk/~dbroadhu/cert/eulmac.gp David
    Message 1 of 8 , Aug 7, 2009
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      --- In primenumbers@yahoogroups.com,
      "Cino Hilliard" <hillcino368@...> wrote:

      > Ok David, I give up.
      >
      > How did you do it?

      http://physics.open.ac.uk/~dbroadhu/cert/eulmac.gp

      David
    • David Broadhurst
      ... f(x,lx) = x/lx - 1/lx + a/lx^2 - b/lx^3; modify at will a=1; b=1; m=2; chosen values p50 terms=10;trunc=10^4;g=f(x,lx);d=[];bv=bernvec(terms+1);
      Message 2 of 8 , Aug 8, 2009
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        --- In primenumbers@yahoogroups.com,
        "Cino Hilliard" <hillcino368@...> wrote:

        > I want to examine
        > sm(m,n)=sum(x=m,n,x/log(x)-1/log(x)+a/log(x)^2-b/log(x)^3)
        > a=1,b=1,m=2

        f(x,lx) = x/lx - 1/lx + a/lx^2 - b/lx^3; \\ modify at will
        a=1; b=1; m=2; \\ chosen values

        \p50
        terms=10;trunc=10^4;g=f(x,lx);d=[];bv=bernvec(terms+1);

        {for(n=1,2*terms-1, \\ store odd derivatives
        g=deriv(g,x)+1/x*deriv(g,lx);if(n%2,d=concat(d,g)));}

        fx(x)=f(x,log(x));
        dx(y)=subst(subst(d,x,y),lx,log(y));

        {emac(m,n)=local(dm,dn);
        dm=dx(m);dn=dx(n);intnum(x=m,n,fx(x))+(fx(n)+fx(m))/2
        +sum(k=1,terms,bv[k+1]/((2*k)!)*(dn[k]-dm[k]));}

        s=sum(k=2,trunc-1,fx(k));

        {sm(m,n) = \\ as requested
        if(n>trunc&&trunc>m&&m>1,s-sum(k=2,m-1,fx(k))+emac(trunc,n));}

        for(k=10,22,print([k,sm(m,sqrtint(10^k))]))

        realprecision = 57 significant digits (50 digits displayed)
        [10, 455051173.78013348859945302791104662093667330309988]
        [11, 4118034570.4638268343263084222227560024582402009128]
        [12, 37607913583.517014991610186636613681414323919745603]
        [13, 346065399465.03248684994266161913298076743032790216]
        [14, 3204941752475.7266213978929589359007763992788467339]
        [15, 29844569450368.625194208305114011697818820369920647]
        [16, 279238341514801.87986838501363369093160129071417051]
        [17, 2623557157209703.2566630532132281009952796686474077]
        [18, 24739954285430064.167128134429584525396645626800130]
        [19, 234057667279245072.15948934429135642246573100181670]
        [20, 2220819602565566257.9962162533153260118661266449421]
        [21, 21127269485065195786.472528704950924409440892097567]
        [22, 201467286689267167012.50527750985870920146256019519]

        Comment: This is an inefficient method of fiddling with Li(x).
        Cino is trying to avoid integration. Yet it is needed for
        Euler-Maclaurin summation.

        David
      • Cino Hilliard
        Hi David, You did a lot of work here to develop emac(m,n).I appreciate it as it saved me a lot of time and possibly money. Thank you. ... Maybe. Anyway I don t
        Message 3 of 8 , Aug 8, 2009
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          Hi David,
          You did a lot of work here to develop emac(m,n).I appreciate
          it as it saved me a lot of time and possibly money.
          Thank you.

          David Broadhurst wrote:
          > Comment: This is an inefficient method of fiddling with Li(x).

          Maybe. Anyway I don't mind inefficiencies as I have to use them
          everyday like the QWERTY keyboard and all the right handed stuff
          around for this lefty.

          Surprizingly, the most efficient combination in the real world for moving a mass from point A to point B with the least expenditure
          of energy, is a human riding a bicycle on a flat surface. Then
          there is the hummingbird which is one of the the least efficient systems that has been around for a while.

          Nevertheless, Your emac(m,n) gets good mileage and is more
          accurate than Li(x) =-eint1(log(1/x)) and
          li(x,n) = lg=log(x);x*sum(k=1,n,(k-1)!/lg^k)

          gp > p23=1925320391606818006727
          gp > em23=emac(2,sqrt(10^23))
          %116 = 1925320391608063705941.14436225628
          gp > Li23=Li(10^23)
          %117 = 1925320391614054155138.78012956636
          gp > R23=R(10^23)
          %118 = 1925320391607837268776.09905063742
          gp > 1-em23/p23
          %122 = -6.4700878855011600657703221947361 E-13
          gp > 1-Li23/p23
          %123 = -3.7584125963269129092185767109212 E-12
          gp > 1-R23/p23
          %124 = -5.2939866712178918250597351383218 E-13

          For 10^22, emac is better than R(x).

          The actual sum of the primes < sqrt(n) ~ Pi(n) is not as
          accurate as emac but it is quite good wit RE below.

          For sum of primes < 10^n, See the b-file at
          http://www.research.att.com/~njas/sequences/A046731

          sump11 = 201467077743744681014
          gp > p22 = 201467286689315906290
          %137 = 201467286689315906290
          gp > 1. - sump11/p22
          %139 = 0.00000103711..
          This is magnitudes better than RE of 10^22/log(10^22)-1) =
          0.000423348356239

          I find this sum of primes <= sqrt(n) ~ primes <= n to be an
          amazing property of prime numbers and the integers.

          Bottomline, thanks to David, we have an eulermac function in
          Pari.

          Cheers and Roebuck,
          Cino
        • David Broadhurst
          ... Glad to help. ... Not so amazing when you realize that the integrals I1(N) = intnum(x=2, sqrt(N), x/log(x)) I2(N) = intnum(y=2, N , 1/log(y)) differ
          Message 4 of 8 , Aug 9, 2009
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            --- In primenumbers@yahoogroups.com,
            "Cino Hilliard" <hillcino368@...> wrote:

            > Hi David,
            > You did a lot of work here to develop emac(m,n).
            > I appreciate it as it saved me a lot of time and
            > possibly money. Thank you.

            Glad to help.

            > I find this sum of primes <= sqrt(n) ~ primes <= n to be
            > an amazing property of prime numbers and the integers.

            Not so amazing when you realize that the integrals

            I1(N) = intnum(x=2, sqrt(N), x/log(x))
            I2(N) = intnum(y=2, N , 1/log(y))

            differ only by a constant, no matter what the value of N.

            Proof: Transform the latter by setting y = x^2. Then
            I2(N) - I1(N) = - I1(2) = 1.92242131492155809316615998937951547...

            David
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