A guide to Tauberian theorems for arithmetic applications
Lillian B. Pierce, Caroline L. Turnage-Butterbaugh, Asif Zaman
TL;DR
The paper develops two precise Tauberian frameworks for nonnegative coefficient Dirichlet series: a Weak version yielding leading-term asymptotics and a Strong version delivering power-saving remainders. It clarifies the linkage between pole data of A(s) and asymptotic growth of partial sums, using Laplace and Mellin-transform techniques, smoothing via approximations to the identity, and contour-shifting to control residues. The work provides rigorous proofs, explicit remainder terms, and counterexamples that delineate the sharpness of hypotheses, with concrete applications to Manin's conjecture, subgroup growth, and counting number fields. Together, these results offer practical, verifiable criteria for obtaining quantitative asymptotics in arithmetic settings and caution about the limits of weaker hypotheses.
Abstract
A Tauberian theorem deduces an asymptotic for the partial sums of a sequence of non-negative real numbers from analytic properties of an associated Dirichlet series. Tauberian theorems appear in a tremendous variety of applications, ranging from well-known classical applications in analytic number theory, to new applications in arithmetic statistics, group theory, and the intersection of number theory and algebraic geometry. The goal of this article is to provide a useful reference for practitioners who wish to apply a Tauberian theorem. We explain the hypotheses and proofs of two types of Tauberian theorems: one with and one without an explicit remainder term. We furthermore provide counterexamples that illuminate that neither theorem can reach an essentially stronger conclusion unless its hypothesis is strengthened.