If we now consider all possible values for `a`

from `1`

upto `x`

, we get:
$\begin{aligned}
|{ (a, b) : ab <= x }|
= \sum_{a=1}^x |{ b: b <= x/a }|
\leq \sum_{a=1}^x |x/a|
\leq x \sum_{a=1}^x (1/a)
\leq x \log x
\end{aligned}$

To show that the harmonic numbers are upper bounded by $\log$,
can integrate: $\sum_{i=1}^n 1/i \leq \int_0^n 1/i = \log n$
#### § Relationship to Euler Mascheroni constant

This is the limit $\gamma \equiv \lim_{n \to \infty} H_n - \log n$. That this is a constant
tells us that these functions grow at the same rate. To see that this si indeed a constant,
consider the two functions:
- $f(n) \equiv H_n - \log n$ which starts at $f(1) = 1$ and strictly decreases.
- $g(n) \equiv H_n - \log(n+1)$ start lower at $g(1) 1 - \log 2$ and strictly increases. [why? ]
- Also, $\lim_n f(n) - g(n) = 0$. So these sandwhich something in between, which is the constant $\gamma$.