Tauberian Theory: A Century of Developments

Tauberian theory is part of summability theory (methods of assigning a value to a divergent series or integrals, usually by taking the limit of some kind of running average value). The general framework of a Tauberian theorem is that, given a summability method A and a summability method B, we seek a theorem that states: If a series or integral is summable with method A, and specified extra hypotheses are true, then it is also summable with method B. The most common case is that method B is ordinary convergence, so we are concluding that a potentially-divergent series that is summable is in fact convergent, provided some conditions are met. The theory was named by G. H. Hardy and J. E. Littlewood in 1913 after the Austrian mathematician Alfred Tauber, who proved the first such result in 1897. Much of the early work was aimed at applications in number theory, and in particular to simplifying proofs of the prime number theorem, but today it has applications in many areas of mathematics.

The present volume is a very erudite monograph on the subject. Although not comprehensive, it does cover all the most important topics. It includes good surveys for the areas it does not cover, and it gives a very good historical view of the whole subject. The first chapter, although devoted to Hardy and Littlewood’s early work, also gives a very good overview of Tauberian theory.

One of the difficulties of Tauberian theory is that there are so many different summation methods, and potentially there would be theorems connecting each pair. The organization of the present book is very good and it manages to avoid overwhelming the reader with choices. The discussion of Wiener’s theory (Chapter II) is especially good; this theory has been reformulated several times, and the book gives both a modern exposition and a good historical development. There is good coverage of both real Tauberians (power series) and complex Tauberians (analytic functions; these are the ones important for prime number theory). The book has a good selection of applications to other areas of mathematics, in particular number theory; there are a half-dozen proofs of the prime number theorem and a proof of the Hardy–Ramanujan strong asymptotic formula for the number of partitions of an integer.

The book is well-written and well-organized, but the information is very dense and so it is not a good introductory text for most people. Many analysis books have some coverage of Tauberian theorems. Two good classical introductions are in Chapter 7 of Titchmarsh’s The Theory of Functions and in Chapter 7 of de Bruijn’s Asymptotic Methods in Analysis, although these only cover real Tauberians. A good look at the Tauberians useful in number theory is in Chapter II.7 of Tenenbaum’s Introduction to Analytic and Probabilistic Number Theory. A thoroughly-modern introduction is in Chapters 1–3 of the recent book of Choimet and Queffélec, Twelve Landmarks of Twentieth-Century Analysis; this covers both real and complex Tauberians.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.