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

$\lim_{r \rightarrow 1-} (1-r) \sum_{n=1}^\infty \frac{n a_n r^n}{1-r^n} = A . \,$

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

Examples

• $\sum_{n=1}^\infty \frac{\mu(n)}{n} = 0 (\mathrm{L})$, 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.