Lambert summation

In mathematical analysis and analytic number theory, Lambert summation is a summability method for summing infinite series related to Lambert series specially relevant in analytic number theory.

Definition

Define the Lambert kernel by $L(x)=\log (1/x)\frac{x}{1-x}$ with $L(1)=1$. Note that $L(x^n)>0$ is decreasing as a function of $n$ when $0<x<1$. A sum $\sum _{n=0}^{\mathrm{\infty }}{a}_n$ is Lambert summable to $A$ if $\underset{x\to 1^{-}}{\lim }\sum _{n=0}^{\mathrm{\infty }}{a}_{n}L(x^n)=A$, written $\sum _{n=0}^{\mathrm{\infty }}{a}_n=0(\mathrm{L})$.

Abelian and Tauberian theorem

Abelian theorem: If a series is convergent to $A$ then it is Lambert summable to $A$.

Tauberian theorem: Suppose that $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ is Lambert summable to $A$. Then it is Abel summable to $A$. In particular, if $\sum _{n=0}^{\mathrm{\infty }}{a}_n$ is Lambert summable to $A$ and $n{a}_n\ge -C$ then $\sum _{n=0}^{\mathrm{\infty }}{a}_n$ converges to $A$.

The Tauberian theorem was first proven by G. H. Hardy and John Edensor Littlewood but it was not independent of number theory, in fact they used a number-theoretic estimate which is somewhat stronger than the prime number theorem itself. The unsatisfactory situation aronund the Lambert Tauberian was resolved by Norbert Wiener.

Examples

• $\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (n)}{n}=0(\mathrm{L})$, where μ is the Möbius function. Hence if this series converges at all, it converges to zero. Note that the sequence $\frac{\mu (n)}{n}$ satisfies the Tauberian condition, therefore the Tauberian theorem implies $\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (n)}{n}=0$ in the oridnary sense. This is equivalent to the prime number theorem.
• $\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{\Lambda }(n)-1}{n}=-2\gamma (\mathrm{L})$ where $\mathrm{\Lambda }$ is von Mangoldt function and $\gamma$ is Euler's constant. By the Tauberian theorem, the ordinary sum converges and in particular converges to $-2\gamma$. This is equivalent to $\psi (x)\sim x$ where $\psi$ is the second Chebyshev function.

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. pp. 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 978-0-521-84903-6. 
• Norbert Wiener (1932). "Tauberian theorems". Ann. of Math. (The Annals of Mathematics, Vol. 33, No. 1) 33 (1): 1–100. doi:10.2307/1968102.