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Re: Highest n, prime count pi(n), pi(n/2)

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  • ianredwood
    ... Many thanks but the vector part is beyond me. How do you know that the bounds are the ones, 5 and 21, that you cite? Can you supply any literary
    Message 1 of 6 , Feb 11, 2010
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      --- In primenumbers@yahoogroups.com, "maximilian_hasler" <maximilian.hasler@...> wrote:
      >
      >
      >
      > --- In primenumbers@yahoogroups.com, "ianredwood" <ianredwood@> wrote:
      > >
      > > Whoops - I've got to say that again.
      > > What's the highest known n for which, for any 7 <= k <= n, pi(k/2) > pi(k) - pi(k/2)?
      > > Is it bigger than 10^(60)?
      > >
      >
      > pi(k/2) > pi(k) - pi(k/2)
      >
      > <=> pi(k/2)*2 > pi(k)
      >
      > The function pi(k/2)*2 - pi(k) tends to infinity
      > in view of the asymptotic behavious of the pi() function
      > so for all k>5 it is nonnegative and for all k>21 it is strictly positive.
      >
      > vector(20,i,(primepi((1+i)\2)*2-primepi(1+i))) /* i+1 to avoid pi(0) */
      > = [-1, -2, 0, -1, 1, 0, 0, 0, 2, 1, 1, 0, 2, 2, 2, 1, 1, 0, 0, 0]
      >
      > vector(99,i,(primepi((1+i)\2)*2-primepi(1+i)))
      > = [-1, -2, 0, -1, 1, 0, 0, 0, 2, 1, 1, 0, 2, 2, 2, 1, 1, 0, 0, 0, 2, 1, 1, 1, 3, 3, 3, 2, 2, 1, 1, 1, 3, 3, 3, 2, 4, 4, 4, 3, 3,
      > 2, 2, 2, 4, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 4, 3, 3, 2, 4, 4, 4, 4, 4, 3, 3, 3, 3, 2, 2, 1, 3, 3, 3, 3, 3, 2, 2, 2, 4, 3, 3, 3, 5,
      > 5, 5, 4, 4, 4, 4, 4, 6, 6, 6, 5, 5, 5, 5]
      >

      Many thanks but the 'vector' part is beyond me. How do you know that the bounds are the ones, 5 and 21, that you cite? Can you supply any literary reference?

      Thankyou.
    • maximilian_hasler
      ... (...) ... It was mainly meant as illustration... among others, of the fact that I don t understand your k = 7 lower limit... Could you first explain that
      Message 2 of 6 , Feb 11, 2010
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        > > --- In primenumbers@yahoogroups.com, "ianredwood" wrote:
        > > >
        > > > Whoops - I've got to say that again.
        > > > What's the highest known n for which,
        > > > for any 7 <= k <= n, pi(k/2) > pi(k) - pi(k/2)?
        > > > Is it bigger than 10^(60)?
        (...)
        > > vector(99,i,(primepi((1+i)\2)*2-primepi(1+i)))
        > > = [-1, -2, 0, -1, 1, 0, 0, 0, 2, 1, 1, 0, 2, 2, 2, 1, 1, 0, 0, 0, 2, 1, 1, 1, 3, 3, 3, 2, 2, 1, 1, 1, 3, 3, 3, 2, 4, 4, 4, 3, 3,
        > > 2, 2, 2, 4, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 4, 3, 3, 2, 4, 4, 4, 4, 4, 3, 3, 3, 3, 2, 2, 1, 3, 3, 3, 3, 3, 2, 2, 2, 4, 3, 3, 3, 5,
        > > 5, 5, 4, 4, 4, 4, 4, 6, 6, 6, 5, 5, 5, 5]
        > >
        > Many thanks but the 'vector' part is beyond me.

        It was mainly meant as illustration...
        among others, of the fact that I don't understand your k >= 7 lower limit... Could you first explain that one ?
        (From the above, one sees that for example for k=21 (position of the last "0" in the list), one has equality

        pi(k/2) = pi(k) - pi(k/2)

        i.e. for this k your initial inequality is not satisfied,
        and one sees that k=5 is the largest value for which (in the domain calculated above), we have strict inequality in the "wrong" sense,

        pi(k/2) < pi(k) - pi(k/2)

        So I don't know what the k >= 7 limit comes from.)

        > How do you know that the bounds are the ones, 5 and 21,
        > that you cite? Can you supply any literary reference?

        no, sorry...
        Hm, at a second look, it seems to follow from the equations in
        http://en.wikipedia.org/wiki/Prime_number_theorem#Bounds_on_the_prime-counting_function

        According to that, the function

        f:= x -> x/ln(x)*(1+1/ln(x));

        is a lower bound, and

        g:= x -> x/ln(x)*(1+1/ln(x)+251/100/ln(x)^2);

        is an upper bound of pi(x), for x large enough, thus:

        D(k) = pi(k/2)*2 - pi(k) > f(k/2)*2 - g(k)

        which yields, if my Maple is correct, something equivalent to
        D(k) ~ ln(2) * k * ln(k)^2 / ln(k/2)^4.

        Can you confirm this calculation ?

        Maximilian
      • Maximilian Hasler
        ... x ≥ 355991 according to the WP page. And I plotted the difference up to 10^6, there s really absolutely no doubt for smaller values ! M. [Non-text
        Message 3 of 6 , Feb 11, 2010
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          > It's the term 'for x large enough' that undoes the claim. How large is large enough?

          x ≥ 355991 according to the WP page.

          And I plotted the difference up to 10^6, there's really absolutely no
          doubt for smaller values !

          M.


          [Non-text portions of this message have been removed]
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