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 $$L(x)=\log(1/x)\frac{x}{1-x}$$ with $$L(1)=1$$. Note that $$L(x^n)>0$$ is decreasing as a function of $$n$$ when $$0<x<1$$. A sum $$\sum_{n=0}^\infty a_n $$ is Lambert summable to $$A$$ if $$\lim_{x\to 1^-}\sum_{n=0}^\infty a_n L(x^n)=A$$, written $$\sum_{n=0}^\infty a_n=A\,\,(\mathrm{L})$$.

Abelian and Tauberian theorem
Abelian theorem: If a series is convergent to $$A$$ then it is Lambert summable to $$A$$.

Tauberian theorem: Suppose that $$\sum_{n=1}^\infty a_n$$ is Lambert summable to $$A$$. Then it is Abel summable to $$A$$. In particular, if $$\sum_{n=0}^\infty a_n$$ is Lambert summable to $$A$$ and $$na_n\geq -C$$ then $$\sum_{n=0}^\infty a_n$$ converges to $$A$$.

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

 * $$\sum_{n=1}^\infty \frac{\mu(n)}{n} = 0 \,(\mathrm{L})$$, where &mu; is the Möbius function. Hence if this series converges at all, it converges to zero. Note that the sequence $$ \frac{\mu(n)}{n}$$ satisfies the Tauberian condition, therefore the Tauberian theorem implies $$\sum_{n=1}^\infty \frac{\mu(n)}{n}=0$$ in the ordinary sense. This is equivalent to the prime number theorem.
 * $$\sum_{n=1}^\infty \frac{\Lambda(n)-1}{n}=-2\gamma\,\,(\mathrm{L})$$ where $$\Lambda$$ is von Mangoldt function and $$\gamma$$ is Euler's constant. By the Tauberian theorem, the ordinary sum converges and in particular converges to $$-2\gamma$$. This is equivalent to $$\psi(x)\sim x$$ where $$\psi$$ is the second Chebyshev function.