In front of my house there is a sidewalk and about 2-3 feet of pavement behind a fence which I'm assuming was formerly dirt that was paved over. I've done my best to work with what I've got and make it look nice, but I'd love to find a solution to the weeds that keep popping up in the cracks in the pavement and sidewalk. 

The front of my house gets a lot of sun but has poor drainage. Most days the area is dry and dusty. The cracks gather a lot of dusty dirt over time and I've pulled out weeds with pretty large root balls. Is there any particular type of groundcover or seed I can sow in the sidewalk cracks or in my single garden square that will outgrow the weeds, survive, and maybe even look good? This is in the Northeast US in zone 6.

Edit: The cracks are 1"-2" wide and always filled with 1"-2" of dirt. Dirt often builds up over the cracks to about 4"+ sizes. I usually try to pull the weeds and sweep it into my sidewalk plot, but it would be much easier/more satisfying to throw some seeds in the sidewalk dirt "chunks" and cracks. I don't have a picture at the moment but I will post one as soon as I can!