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 ${\displaystyle \sum a_{n}=A\,(\mathrm {L} )}$, if

${\displaystyle \lim _{r\rightarrow 1-}(1-r)\sum _{n=1}^{\infty }{\frac {na_{n}r^{n}}{1-r^{n}}}=A.\,}$

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

Examples

• ${\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}=0(\mathrm {L} )}$, where μ is the Möbius function. Hence if this series converges at all, it converges to zero.

See also

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

References

• Jacob Korevaar (2004). Tauberian theory. A century of developments. Grundlehren der Mathematischen Wissenschaften. 329. Springer-Verlag. p. 18. ISBN 3-540-21058-X.
• Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. 97. Cambridge: Cambridge Univ. Press. pp. 159–160. ISBN 0-521-84903-9.
• Norbert Wiener (1932). "Tauberian theorems". Ann. of Math. The Annals of Mathematics, Vol. 33, No. 1. 33 (1): 1–100. doi:10.2307/1968102. JSTOR 1968102.