Tenenbaum's book is indeed one of the best on the subject; it's well-motivated and quite accessible. If you go a bit further back, there are also the Probabilistic Number Theory books by P. D. T. A. Elliot; volume I is on Mean Value Theorems, while volume II is on Central Limit Theorems. These are both a bit more specialised and slightly outdated. Even older still is Probabilistic Methods In the Theory of Numbers by J. Kubilius.