§ The number of pairs
Fix a given
(a,b) such that
ab ≤ x implies that
b ≤ x/a, or there are only
x/a possible values for
If we now consider all possible values for
x, we get:
To show that the harmonic numbers are upper bounded by ,
§ Relationship to Euler Mascheroni constant
This is the limit . 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:
- which starts at and strictly decreases.
- start lower at and strictly increases. [why? ]
- Also, . So these sandwhich something in between, which is the constant .