Tauberian theorems

Tauberian theorems are a set of results in mathematical analysis, particularly in the field of summability and asymptotic analysis. They provide conditions under which certain types of series or transforms can be inferred from the behavior of their generating functions or sequences. The general idea is to connect the asymptotic behavior of a sequence or a series with conditions imposed on its transform, such as the Laplace transform or the Dirichlet series.

Abelian and Tauberian theorems are concepts from mathematical analysis and number theory, specifically related to the convergence of series and the properties of generating functions. Here’s a brief overview of each: ### Abelian Theorem The Abelian theorem typically refers to the Abel's test, which is a criterion for the convergence of series and power series.

Haar's Tauberian theorem is a result in the field of analytic number theory and harmonic analysis, specifically dealing with summability methods and their connection to convergence of series. The theorem is named after mathematician Alfréd Haar. The basic idea behind Haar's theorem is to establish conditions under which the summation of an infinite series can be deduced from information about the behavior of its partial sums.

The Hardy–Littlewood Tauberian theorem is an important result in analytic number theory and summability theory. It provides a bridge between the growth conditions of a generating function and the convergence behavior of its associated series. In particular, it establishes conditions under which the summation of a series can be related to the growth of its generating function.

Littlewood's Tauberian theorem is a result in the field of mathematical analysis that connects the properties of series (or sequences) and their associated generating functions, specifically in the context of summability methods. The theorem provides conditions under which the convergence of a series can be inferred from the behavior of its generating function, particularly in relation to its analytic properties.

A "slowly varying function" is a concept from asymptotic analysis and number theory that refers to a function that grows very slowly compared to a linear function as its argument tends to infinity. More formally, a function \( L(x) \) is said to be slowly varying at infinity if: \[ \lim_{x \to \infty} \frac{L(tx)}{L(x)} = 1 \] for all \( t > 0 \).

The Wiener–Ikehara theorem is a result in analytic number theory, which deals with the asymptotic distribution of the partition function \( p(n) \), specifically in relation to the number of partitions of an integer. More formally, it connects the asymptotic behavior of a certain generating function with the distribution of partitions.