# Lambert summation method

2010 Mathematics Subject Classification: Primary: 40C [MSN][ZBL]

A summation method for summing series of complex numbers which assigns a sum to certain divergent series: it is regular in that it assigns the sum in the usual sense to any convergent series (an Abelian theorem). The series $$\sum_{n=1}^\infty a_n$$ is summable by Lambert's method to the number $A$, written ${} = A \ \mathrm{(L)}$ if $$\lim_{y \searrow 0} F(y) = A$$ where $F(y)$ is a Lambert series in $\exp(-y)$: $$F(y) = \sum_{n=1}^\infty a_n \frac{n y \exp(-ny)}{1-\exp(-ny)}$$ for $y>0$, if the series on the right-hand side converges. The method was proposed by J.H. Lambert [1]. The summability of a series by the Cesàro summation method $(C,k)$ for some $k > -1$ (cf. Cesàro summation methods) to the sum $A$ implies its summability by the Lambert method to the same sum, and if the series is summable by the Lambert method to the sum $A$, then it is also summable by the Abel summation method to the same sum.

As an example, $$\sum_{n=0}^\infty \frac{\mu(n)}{n} = 0\ \mathrm{(L)}$$ where $\mu$ is the Möbius function. Hence if this series converges at all, it converges to zero.

#### References

 [1] J.H. Lambert, "Anlage zur Architektonik" , 2 , Riga (1771) [2] G.H. Hardy, "Divergent series" , Clarendon Press (1949) [3] Jacob Korevaar (2004). "Tauberian theory. A century of developments". Grundlehren der Mathematischen Wissenschaften 329. Springer-Verlag (2004). ISBN 3-540-21058-X. p. 18. [4] Hugh L. Montgomery; Robert C. Vaughan (2007). "Multiplicative number theory I. Classical theory". Cambridge tracts in advanced mathematics 97. Cambridge: Cambridge Univ. Press (2007). ISBN 0-521-84903-9. pp. 159–160. [5] Norbert Wiener "Tauberian theorems". Ann. Of Math. 33 (1932) 1–100. DOI 10.2307/1968102. JSTOR 1968102.
How to Cite This Entry:
Lambert summation method. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Lambert_summation_method&oldid=33030
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article