How can the prime number theorem be used to understand this proof?

[~Death~]

could someone please help me understand this proof given in an article by William Miller

(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

 

[hamster143]

pi(t) jumps by 1 when t is prime. Therefore log(t) d(pi(t)) contributes log(t) for prime integers and 0 for all other values of t.