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Could someone please help me understand this proof given in an article by William Miller. It is supposed to follow from the prime number theorem that given $A(x)$ the sum of all primes less than or equal to $x$ and $\theta(x)$ the sum of the logarithm of all primes less than or equal to $x$,

$$A(x)\sim \frac{x^2}{2\log x} \ \ \ \rm and \ \ \ \theta(x) \sim x,$$

the following identity is used:

$$\theta(x) = \int_1^x \log(t)\mathrm{d}(\pi(t)),$$

where $\pi(t)$ is the prime counting function. I don't understand why this is. Here $\sim$ means asymptotic to i.e. $\lim_{n\to\infty} \frac{f(x)}{g(x)}=1$.

  • 2
    Could you give a link to this article you speak of, please?2010-09-26
  • 0
    Maybe you are interested in [this](http://math.stackexchange.com/q/115230/19341)?2012-03-16

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