Explore BrainMass

Explore BrainMass

    Logarithmic integral: two forms

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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.

    © BrainMass Inc. brainmass.com September 27, 2022, 4:32 pm ad1c9bdddf
    https://brainmass.com/math/number-theory/617265

    Attachments

    SOLUTION This solution is FREE courtesy of BrainMass!

    Please see the attached .pdf file for the solution.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com September 27, 2022, 4:32 pm ad1c9bdddf>
    https://brainmass.com/math/number-theory/617265

    Attachments

    ADVERTISEMENT