0
$\begingroup$

Let $$A(x)=x\sum_{p \leq x} 1, B(x)=\frac{3}{5}\sum_{x

  • 3
    Where does $B(x)$ comes from? The $\frac 35$ is really mesmerizing.2011-12-11
  • 0
    I am also puzzled about that $\frac{3}{5}$.2011-12-11

1 Answers 1

3

The sum of the primes up to $x$ is, asymptotically, $(1/2)x^2/\log x$. So asymptotically, $$B(x)=(3/5)((1/2)(2x)^2/\log(2x)-(1/2)x^2/\log x))$$ Looks like about $(9/10)x^2/\log x$.

  • 0
    Where do you get that asymptotic formula for the sum of primes up to x? Can I find this result in Apostol?2011-12-11
  • 0
    You can get ot out of http://oeis.org/A007504 but there are probably more direct ways. Maybe a better place is http://mathworld.wolfram.com/PrimeSums.html2011-12-11
  • 1
    @Rob, just try to apply summation by part and use prime number theorem, you can get the asymptote directly.2011-12-11