Reprsentations of Natural Numbers as the Sum of a Prime and a $k$-th Power

Abstract

Subject to the Riemann hypothesis for Dirichlet $L$ functions an asymptotic formula is obtained for the number of representations of a natural number $n$ as the sum of a prime and a $k$-th power, valid for almost $n$. Estimates for the error term in the asymptotic formula as well as for the size of the exceptional set are of a smaller order of magnitude than was known previously.