Lambert summation method

2020 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.