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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


    (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.

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    Solution Summary

    The two forms of the logarithmic integral are derived. Also, the first form is used to show that the logarithmic integral li(x) has the same asymptotic behavior as the function pi(x), where pi(x) is the prime-counting function.