Hardy–Littlewood Tauberian theorem

In mathematical analysis, the Hardy–Littlewood Tauberian theorem is an important idea that helps us understand patterns in numbers. It connects two different ways of looking at a series, which is a list of numbers added together.

The theorem says that if one way of adding the numbers shows a certain pattern as we get closer to a special point, then another way of adding the numbers will show a related pattern as the list gets longer and longer.

This theorem was proved in 1914 by two mathematicians, G. H. Hardy and J. E. Littlewood. Later, in 1930, another mathematician named Jovan Karamata found a simpler way to prove it. This work helps mathematicians understand how different methods of adding numbers can give us clues about the overall behavior of those numbers.

Statement of the theorem

Series formulation

This part of the theorem talks about numbers that are always zero or more. If we add up these numbers in a special way using a math trick, and the result looks like 1/y when y gets very small, then another way of adding them up will look like n when n gets very big.

Integral formulation

This part uses a different math idea. It looks at a special kind of math function and how it changes. The theorem says that if one part of this function behaves in a certain way when numbers get very small, then another part will behave in a matching way when numbers get very big. This helps us understand how these functions grow and change over time.

Karamata's proof

Karamata found a simple way to prove the Hardy–Littlewood Tauberian theorem. He looked at special math functions and showed that certain patterns hold true. This helped connect two different ways of looking at series in math.

In 1911, Littlewood proved something new using ideas from Abel. He showed that if a certain condition is met, then the sum of a series stays the same. This was an important step before the Hardy–Littlewood Tauberian theorem.

In 1915, Hardy and Littlewood used their theorem to help prove the prime number theorem. They showed that the way certain numbers add up relates to how often prime numbers appear. This was a big discovery in number theory.