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

Given a Dirichlet series L(s) = ∑ aₙ n⁻ˢ, the asymptotic growth rate of ∑n≤ X aₙ 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 with error terms depending only on the average size of $L(σ+it)$ as t varies, and we take care to track explicit dependence on various parameters. This often leads to stronger error bounds, and introduces strong connections between asymptotic counting problems and moments of L-functions. We provide self-contained statements of Tauberian theorems in anticipation that these results can be used ``out of the box'' to prove new asymptotic expansions. We demonstrate this by proving square root saving error bounds for the number of Cₙ-extensions of Q of bounded discriminant when $n=3$, $4$, $8$, $16$, or $2p$ for p an odd prime.