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# Logarithmic integral: two forms

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Define the logarithmic integral li(x) as the integral of the function 1/(log t) from t = 2 to t = x, where x > 2 and "log" denotes the natural logarithm.

(a) Determine constants A and B such that li(x) can be expressed in the following two forms:

(i) li(x) = x/(log x) + A + g(x), where g(x) is the integral of the function 1/[(log t)^2] from t = 2 to t = x

(ii) li(x) = x/(log x) + x/[(log x)^2] + B + 2h(x), where h(x) is the integral of the function 1/[(log t)^3] from t = 2 to t = x

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(b) Use the form of li(x) given in (a)(i) above to prove that li(x) ~ x/(log x).

Deduce that the Prime Number Theorem can be expressed in the form pi(x) ~ li(x), where (for any real number x) pi(x) is the prime-counting function that gives the number of primes less than or equal to x.