On a Tauberian Theorem of Ingham and Euler-Maclaurin Summation

TL;DR

This paper analyzes and clarifies the conditions under which a Tauberian theorem of Ingham and Euler-Maclaurin summation can be applied in analytic number theory, providing generalizations for complex and multi-dimensional series.

Contribution

It rigorously examines the technical conditions of Ingham's Tauberian theorem and Euler-Maclaurin summation, ensuring their correct application and extending their scope to complex and multi-dimensional series.

Findings

Technical conditions are satisfied in recent applications

Generalized Euler-Maclaurin for complex and multi-dimensional series

Confirmed the validity of asymptotic formulas in key cases

Abstract

We discuss two theorems in analytic number theory and combinatory analysis that have seen increased use in recent years. A corollary to a Tauberian theorem of Ingham allows one to quickly prove asymptotic formulas for arithmetic sequences, so long as the corresponding generating function exhibits exponential growth of a certain form near its radius of convergence. Two common methods for proving the required analytic behavior are modular transformations and Euler-Maclaurin summation. However, these results are sometimes stated without certain technical conditions that are necessary for the complex analytic techniques that appear in Ingham's proof. We carefully examine the precise statements and proofs of these results, and find that in practice, the technical conditions are satisfied for those cases appearing in recent applications. We also generalize the classical approach of Euler-Maclaurin summation in order to prove asymptotic expansions for series with complex values, simple poles, or multi-dimensional summation indices.