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.

我们讨论了解析数论和组合分析中的两个定理，这些定理近年来已得到越来越多的使用。对英厄姆的陶伯定理的推论使人们能够快速证明算术序列的渐近公式，只要相应的生成函数在其收敛半径附近表现出某种形式的指数增长即可。证明所需分析行为的两种常用方法是模数转换和Euler-Maclaurin求和。但是，有时会在没有特定技术条件的情况下陈述这些结果，而这些条件是Ingham证明中出现的复杂分析技术所必需的。我们仔细检查了这些结果的精确陈述和证据，发现在实践中，满足了最近申请中出现的那些情况的技术条件。