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


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