On a Tauberian theorem of Ingham and Euler-Maclaurin summation - Kölner UniversitätsPublikationsServer

Bringmann, Kathrin, Jennings-Shaffer, Chris and Mahlburg, Karl . On a Tauberian theorem of Ingham and Euler-Maclaurin summation. Ramanujan J.. DORDRECHT: SPRINGER. ISSN 1572-9303

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DOI:10.1007/s11139-020-00377-5

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.

Item Type:

Journal Article

Creators:

Creators	Email	ORCID	ORCID Put Code
Bringmann, Kathrin	UNSPECIFIED	UNSPECIFIED	UNSPECIFIED
Jennings-Shaffer, Chris	UNSPECIFIED	UNSPECIFIED	UNSPECIFIED
Mahlburg, Karl	UNSPECIFIED	UNSPECIFIED	UNSPECIFIED