I'm the third owner of an Audi A3 2001 1.9TDI with 5-gear manual transmission. I'd refer to the owners manual but I didn't get it when I bought the car, someone must have misplaced it. I can't find much information online.
We didn't get to enjoy the A3 diesel in the states, but every other Audi manual trans from that era that I can look up uses:
Synthetic SAE 75w/90 Gear Oil
Audi Part Number G 005 000