I am planning to buy a Canon 5d mark III, for landscape, wild life and bird photography. What would be an ideal lens with a budget of 1500-2000 dollars? And specifically, what is your opinion on the Canon EF 100-400mm f/5.6?
AT 400mm, where you would use the EF 100-400mm f/4-5.6L the most, the EF 400mm f/5.6 is noticeably sharper, especially at mid frame and on the edges. It also costs about $300 less. You give up Image Stabilization (IS) and the ability to zoom out to 100mm.
For birding and wildlife, IS is not a huge consideration because you're going to need to use fairly fast shutter speeds to freeze the motion of your subjects anyway. IS only helps reduce camera motion. It does nothing for subject motion.
For most typical landscape photos, 100mm is still a little long, so you're likely going to need another lens for that role regardless of what you choose for your long telephoto birding/wildlife lens. Unfortunately, there's not a real good FF standard zoom lens for the $660 left in your budget if you were to purchase the EF 400mm f/5.6L, much less the $300 left in your budget if you purchase the EF 100-400mm f/4-5.6L IS. The closest would be the EF 24-105mm f/4L IS that runs around $1,150.
The 100-400 is a good lens to get started with and in your budget.
I've talked about this configuration in the past -- try here: What size lens is recommended for flying bird photography?