An Explicit Tauberian Theorem taking Averaged Inputs with an Application to Counting Abelian Number Fields

Given a Dirichlet series $L(s) = \sum a_n n^{-s}$, the asymptotic growth rate of $\sum_{n\le X} a_n$ can be determined by a Tauberian theorem. Bounds on the error term are typically controlled by the size of $|L(σ+it)|$ for fixed real part $σ$. We modify this approach to prove new Tauberian theorems