How much do real-estate prices correlate with apartment rental prices? I've been trying to find sources that mention this, and have tried to find graphs of data so I can compare myself, but I haven't been able to. Do real-estate sale prices correlate positively, negatively, or not at all with apartment rental prices? A more nuanced answer than that is also appreciated.
In theory there should be a direct positive correlation (especially in countries where house prices are a part of the index which is used to define rental rates), but in reality there is a lot of variation in the results for this question depending on the country, possible effects of different laws, the business cycle, whether new housing is being built or not, etc.

For Italy, Manganelli, Morano and Tajan (2014) in "House prices and rents. The Italian experience" study response functions and conclude:

2)  Rents depend on real-estate prices, the short-term interest rate and the differences between actual and expected GDP with a one year delayed reaction. 

On the other hand, Ambrose, Eichholtz and Lindenthal (2013) in House Prices and Fundamentals: 355 years of evidence, Journal of Money, Credit and Banking find that when housing prices stray from equilibrium growth, rental rates do not react the same way even if it takes decades for the prices to return to steady growth.

For a recent study on U.S. data (model built to match trends from 1960-2013), I'd check Gelain and Lansing (2014): House prices, expectations, and time-varying fundamentals, Journal of Empirical Finance