Define the Lambert kernel by with . Note that is decreasing as a function of when . A sum is Lambert summable to if , written .
Abelian and Tauberian theorem
Abelian theorem: If a series is convergent to then it is Lambert summable to .
Tauberian theorem: Suppose that is Lambert summable to . Then it is Abel summable to . In particular, if is Lambert summable to and then converges to .
The Tauberian theorem was first proven by
G. H. Hardy and
John Edensor Littlewood but was not independent of number theory, in fact they used a number-theoretic estimate which is somewhat stronger than the prime number theorem itself. The unsatisfactory situation around the Lambert Tauberian theorem was resolved by
Norbert Wiener.
Examples
, where μ is the
Möbius function. Hence if this series converges at all, it converges to zero. Note that the sequence satisfies the Tauberian condition, therefore the Tauberian theorem implies in the ordinary sense. This is equivalent to the
prime number theorem.
Jacob Korevaar (2004). Tauberian theory. A century of developments. Grundlehren der Mathematischen Wissenschaften. Vol. 329.
Springer-Verlag. p. 18.
ISBN3-540-21058-X.