Lambert summation
In mathematical analysis and analytic number theory, Lambert summation is a summability method for summing infinite series related to Lambert series specially relevant in analytic number theory.
Definition
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.
- where is von Mangoldt function and is Euler's constant. By the Tauberian theorem, the ordinary sum converges and in particular converges to. This is equivalent to where is the second Chebyshev function.