Asymptotic analysis of restricted integer partitions

In this talk, we discuss the asymptotic behavior of the number of partitions into a fixed subset of positive integers. The main focus of the talk will be when this subset consists of primes concerning a Chebotarev condition. In special cases, this reduces to partitions into primes in arithmetic progressions. While the study for ordinary partitions goes back to Hardy and Ramanujan's circle method in 1918, partitions into primes were recently re-visited by Vaughan about a decade ago. We give a sharp error term and improve Vaughan's estimates. As an application, we touch upon the monotonicity of this partition function via an asymptotic formula in connection to a result of Bateman and Erd\H{o}s.