Tauberian Theorems on Dirichlet's Series

The object of this paper is to prove that these theorems are true even when r > 1. The argument used is similar to that used in Theorem 22 of the Cambridge Tract on The General Theory of Dirichlet's Series by Hardy and Riesz. Some remarks are also made in the concluding section about the applications of these Tauberian theorems to obtain certain precise results on the abcissae of summability of Dirichlet's series. These results have been anticipated by Ananda Rau.