- http://primes.utm.edu/howmany.shtml

Consequence One: You can Approximate pi(x) with x/(log x - 1)

http://en.wikipedia.org/wiki/Prime-counting_function

http://en.wikipedia.org/wiki/Logarithmic_integral

So why do we use integral(dt/log_e(t))

instead of integral(dt/(log_e(t) - 1)

as the approximation to pi(x)?

Kermit - --- In primenumbers@yahoogroups.com,

Kermit Rose <kermit@...> wrote:

> You can Approximate pi(x) with x/(log x - 1)

Exercise: Find the best value of c such that the derivative

> So why do we use integral(dt/log_e(t))

of x/(log(x)-c) approximates 1/log(x) when log(x) is large.

Solution: The derivative x/(log(x)-c) is

1/(log(x)-c) - 1/(log(x)-c)^2

and expanding in 1/log(x) we obtain

1/log(x) + (c-1)/log(x)^2 + O(1/log(x)^3)

so the best approximation is with c = 1.

David