1. ## Confusing integral question

The question:
Given a positive real number x, let $\displaystyle \pi(x)$ denote the number of primes less than or equal to x. The function Li with domain $\displaystyle (1, \infty)$ is given by:

$\displaystyle Li(x) = \int_{2}^{x} \frac{1}{ln(t)} dt$

and is known as the 'logarithmic integral function'. It has the propery that

$\displaystyle \frac{Li(x)}{\pi(x)} \approx 1$

when x is sufficiently large.

a) Evaluate $\displaystyle \pi(10), \pi(20), \pi(\pi)$
b) Suppose that x > 0. What does $\displaystyle \frac{\pi(x)}{x}$ represent?

My attempt:
a) I just worked out these values in my head. 4, 8 and 2. I was going to take the integral and work it out, but apparently 1/log(x) isn't trivial to integrate...
b) This one I'm not sure about, and there's no answer in my text. My guess is that it's the number of primes per integer, but that could be my ignorance talking.

There's more parts to this question, but I'm going to see if I can attempt them first. Meanwhile, is what I've done the correct approach? Thanks.

Ok, this is the other question I need assistance with:

d) By applying the Mean Value Theorem to Li on the interval $\displaystyle [2, 10^6]$, find a lower bound for $\displaystyle Li(10^6)$

2. Originally Posted by Glitch
The question:
Given a positive real number x, let $\displaystyle \pi(x)$ denote the number of primes less than or equal to x. The function Li with domain $\displaystyle (1, \infty)$ is given by:

$\displaystyle Li(x) = \int_{2}^{x} \frac{1}{ln(t)} dt$

and is known as the 'logarithmic integral function'. It has the propery that

$\displaystyle \frac{Li(x)}{\pi(x)} \approx 1$

when x is sufficiently large.

a) Evaluate $\displaystyle \pi(10), \pi(20), \pi(\pi)$
b) Suppose that x > 0. What does $\displaystyle \frac{\pi(x)}{x}$ represent?

My attempt:
a) I just worked out these values in my head. 4, 8 and 2. I was going to take the integral and work it out, but apparently 1/log(x) isn't trivial to integrate...
Yes, that's correct. Notice that it says that Li and $\pi$ are approximately the same for sufficiently large x. That does not help you at all in finding $\displaystyle \pi(10$, $\displaystyle \pi(20)$, $\displaystyle \pi(\pi)$.

b) This one I'm not sure about, and there's no answer in my text. My guess is that it's the number of primes per integer, but that could be my ignorance talking.
$\displaystyle \pi(x)$ is the number of primes less than x. $\displaystyle \pi(x)/x$, then, is the proportion of integers less than x that are primes.

There's more parts to this question, but I'm going to see if I can attempt them first. Meanwhile, is what I've done the correct approach? Thanks.

Ok, this is the other question I need assistance with:

d) By applying the Mean Value Theorem to Li on the interval $\displaystyle [2, 10^6]$, find a lower bound for $\displaystyle Li(10^6)$
The mean value theorem says that (f(b)- f(a))/(b- a)= f'(c) for some number c, between a and b. Applying that to the function Li(x) on the interval $\displaystyle [2, 10^6]$ says that $\displaystyle \frac{Li(10^6)- Li(2)}{10^6}= Li'(c)$ for some c between 2 and $\displaystyle 10^6$. Of course, Li(2)= 0 and I suspect you can approximate $\displaystyle 10^6- 2$ by $\displaystyle 10^6$ without significant error. So you are saying that $\displaystyle Li(10^6)= 10^6 Li'(c)$ for some c less than $\displaystyle 10^6$. It is, of course, easy to find Li'(x). How large does that get to be?