Difference between the definitions regarding distribution of prime numbers

Question

Following are the two theorems that Hardy and Wright state in their book

Theorem A: The number of primes not exceeding $x$ is given by $\pi(x) \sim \frac{x}{\log{x}}$.

Theorem B: The order of magnitude of $\pi(x)$ is $\pi(x) \asymp \frac{x}{\log{x}}$.

where,

1. $f \sim \phi$ iff $\; \frac{f}{\phi} \to 1\;$, ie. two the two functions are asymptotically similar.
2. $f \asymp \phi$ iff $\;A^{'}\phi < f < A \phi\;$ ie. $\;f\;$ is of same order of magnitude as $\phi$ ( $A$ and $A^{'}$ are constants ).

How are the two theorems different ? Am I missing something trivial ?

Answer 1

They are slightly different. The first gives a limit asymptotic constant equal to $1$, but no bracketing. The second doesn't give a single asymptotic constant, but a bracketing (for $x$ large enough). Obviously, $A'\le1\le A$.

Answer 2

As a concrete example think of f:x -> (sin x) + 2; and g:x -> 2; Its clear that f/g does not converge anywhere, ie f !∼ g, but they are of the same order of magnitude as 0.4g < f < 2g.

In general 2. does not imply 1. x and 2x are a trivial example 2x/x is fixed at 2 but they clearly have the same magnitude.

1 does not imply 2 either in the general case (think of a function ϕ that approaches 0 at some place prior to its asymptotic behaviour where f is 1) but must imply it for some region (x,infinity)