It seems to go up with population somehow, but are there any particular numbers? I can't seem to find this information anywhere.
A city requires 1 amenity for every 2 citizens, starting with its 3rd pop.
AmmenitiesNeeded = mathF.Floor((Population - 2) / 2 );
Here is the full write-up concerning Amenities from the Civ 6 Manual.
This panel displays how happy your population is, as well as a breakdown of any Amenities. Happiness measures the contentment of your people in a single city, and is measured by seeing if the Population of that city has as many Amenities as expected. 1 Amenity is required for each 2 Population, starting at a Population of 3. A content city will not have any modifier applied to it, but a happy city will have a 10% growth increase and a 5% yield increase. Furthermore, an Ecstatic city will have a 20% growth increase and 10% yield increase. Likewise, having too few Amenities will create displeasure among your Citizens, and the growth and yields will receive negative modifiers. At the level of unrest, your growth will stop and you may find rebel partisans in your city.
Amenities can come from a wide variety of sources. Each Luxury resource can contribute 1 Amenity to a city. The number of cities it provides this bonus to vary based on the era for the resource. The Arena, built in the Entertainment Complex, provides Amenities to its local city. The Zoo and Stadium also provide Amenities, but regionally. Some wonders also provide Amenities. Furthermore, there are a few Religious Beliefs and policies that provide Amenities.