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

## TL;DR

This paper develops new Tauberian theorems that relate the asymptotic growth of arithmetic functions to averaged bounds on associated L-functions, leading to improved error estimates in counting number fields.

## Contribution

It introduces explicit Tauberian theorems based on averaged L-function bounds, enhancing error control and connecting asymptotic counting with moments of L-functions.

## Key findings

- Established square root saving error bounds for counting certain number fields.
- Connected asymptotic counting problems with moments of L-functions.
- Provided self-contained Tauberian theorems for broader application.

## Abstract

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 $|L(\sigma+it)|$ for fixed real part $\sigma$. We modify this approach to prove new Tauberian theorems with error terms depending only on the average size of $L(\sigma+it)$ 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 $C_n$-extensions of $\mathbb{Q}$ of bounded discriminant when $n=3$, $4$, $8$, $16$, or $2p$ for $p$ an odd prime.