(This exercise is for people who have taken some calculus.) The Prime Number Theorem says...

 (This exercise is for people who have taken some calculus.) The Prime Number Theorem says that the counting function for primes, π(x), is approximately equal to x/ ln(x) when x is large. It turns out that π(x) is even closer to the value of the definite integral

(a) Show that

This means that and x/ ln(x) are approximately the same when x is large. [Hint. Use L’Hˆopital’s rule and the Second Fundamental Theorem of Calculus.]

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(b) It can be shown that

Use this series to compute numerically the value of for x = 10, 100, 1000, 104, 106, and 109. Compare the values you get with the values of π(x) and x/ ln(x) given in the table on page 92. Which is closer to π(x), the integral or the function x/ ln(x)? (This problem can be done with a simple calculator, but you’ll probably prefer to use a computer or programmable calculator.)

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(c) Differentiate the series in (b) and show that the derivative is actually equal to 1/ ln(t). [Hint. Use the series for ex.]