Mathematicians and scientists do things in similar ways...........



B) π(x)≈Li(x)=∫x21lntdt

Both of those approximations are "good" in the sense that the ratio between the left side and the right side tends to 1 as x gets larger and larger. However, they are not equally good: B) is much better than A), however A) is slightly easier to remember and, with some preparation, you can use it in your head or with pencil and paper for some really large values of x.

Let's take an example. Suppose you want to know how many primes there are less than one trillion (that's 1012). To use method A), we need to know what ln(1012) is. Well, that's obviously 12 times ln(10), and ln(10) is a number you may want to memorize (an approximation for) if you care about prime counting. It's about 2.30. So 12 times this is about 27.6, and therefore the number of primes up to a trillion is about a trillion divided by 27. In other words: roughly 1 number out of 27 in this range is a prime. That's actually a pretty useful way to understand the answer - much more intuitive than the precise number of such primes (which, incidentally, is 37,607,912,018).