Lambert Summable

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

Abelian and Tauberian theorem

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

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

* $\sum_^\infty \frac = 0 \,(\mathrm)$, where μ is the Möbius function. Hence if this series converges at all, it converges to zero. Note that the sequence $\frac$ satisfies the Tauberian condition, therefore the Tauberian theorem implies $\sum_^\infty \frac=0$ in the ordinary sense. This is equivalent to the prime number theorem.
* $\sum_^\infty \frac=-2\gamma\,\,(\mathrm)$ 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.

See also

* Lambert series
* Abel–Plana formula
* Abelian and tauberian theorems

References

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Series (mathematics)
Summability methods

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