Tauberian theorems

Tauberian theorems are mathematical results that connect the behavior of a function when viewed in a transformed form (like a series or integral) to its original properties. Specifically, they help us deduce long-term or average behavior of a sequence or function based on information from its transformed version, often under certain conditions. Think of it as a way to interpret how a process behaves over time by analyzing its summarized or "smoothed" version, bridging the gap between simplified data and the detailed reality. These theorems are fundamental in analysis and number theory for understanding asymptotic properties.