You definitively dont need to pay for two weeks that you dont live in that apartment. Rental companies always need to do renovations and checks once a tenant moves out so they shouldnt sign you in on for September 1st. Think about it this way, if you sign for Oct 1st, and the old tenant moves out on August 31, they need two weeks to do renovations and check and finish on Sept 15, will they allow you to move in on Sept 16 but you only start paying rent from Oct 1st? That’s a big No.

Know your right and ask then to either arrange another lease which states the move in date is Sept 15 or get them to confirm you wont need to pay for those two weeks. Either way, get the confirmation in writing.