could someone please help me understand this proof given in an article by William Miller(adsbygoogle = window.adsbygoogle || []).push({});

(attatched)

its supposed to follow from the prime number theorem that given,

A(x) which is the sum of all primes less than or equal to x

and theta(x) which is the sum of the log of all primes less than or equal to x

A(x) ~ x^2/(2logx) and theta(x) ~ x

the following identity is used, theta(x) = integral from 1 to x of log(t)d(pi(t))

where pi(t) is the prime counting function. I don't understand why this is.

Here ~ means asymptotic to i.e. lim n->infinity f(x)/g(x)=1

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# The prime number theorem

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