Approximation to pi(x)
Consequence One: You can Approximate pi(x) with x/(log x - 1)
So why do we use integral(dt/log_e(t))
instead of integral(dt/(log_e(t) - 1)
as the approximation to pi(x)?
- --- In email@example.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.