I wonder why your doctor would say that. Based on your LMP of August 23, the most likely date you'd have ovulated would have been September 6th. In which case, Jack would be the father. Sperm can live inside you for up to 5 days. An egg lives for a day. By the 13th of September it is unlikely you would still be ovulating or even would have begun to ovulate then.
I am curious though why you didn't trust what your doctor said and sought out the opinions of strangers...
Based on the information given, Jack is the father.