Lambert summation

In mathematical analysis, Lambert summation is a summability method for a class of divergent series.

Definition

A series $sum\; a\_n$ is "Lambert summable" to "A", written $sum\; a\_n\; =\; A\; (mathrm\{L\})$, if

:$lim\_\{r\; ightarrow\; 1-\}\; (1-r)\; sum\_\{n=1\}^infty\; frac\{n\; a\_n\; r^n\}\{1-r^n\}\; =\; A\; .\; ,$

If a series is convergent to "A" then it is Lambert summable to "A" (an Abelian theorem).

Examples

* $sum\_\{n=0\}^infty\; frac\{mu(n)\}\{n\}\; =\; 0\; (mathrm\{L\})$, where μ is the Möbius function. Hence if this series converges at all, it converges to zero.

ee also

* Lambert series
* Abelian and tauberian theorems

References

*
*cite book | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) | coauthors=Robert C. Vaughan | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=0-521-84903-9 | pages=159-160
*cite journal | author=Norbert Wiener | authorlink=Norbert Wiener | title=Tauberian theorems | journal=Ann. Of Math. | year=1932 | volume=33 | pages=1–100 | doi=10.2307/1968102