# Lambert summation method

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)$: '"UNIQ-MathJax4-QINU"' for$y>0$, if the series on the right-hand side converges. The method was proposed by J.H. Lambert [[#References|[1]]]. The summability of a series by the Cesàro summation method$(C,k)$for some$k > -1$(cf. [[Cesàro summation methods|Cesàro summation methods]]) to the sum$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$, then it is also summable by the [[Abel summation method|Abel summation method]] to the same sum. As an example, '"UNIQ-MathJax5-QINU"' where$\mu\$ is the Möbius function. Hence if this series converges at all, it converges to zero.