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 $\sum_{n = 0}^{\infty} a_{n}$ is Lambert summable to $A$ if $lim_{x \to 1^{-}} \sum_{n = 0}^{\infty} a_{n} L (x^{n}) = A$ , written $\sum_{n = 0}^{\infty} a_{n} = A (L)$ .

Abelian and Tauberian theorem

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

Tauberian theorem: Suppose that $\sum_{n = 1}^{\infty} a_{n}$ is Lambert summable to $A$ . Then it is Abel summable to $A$ . In particular, if $\sum_{n = 0}^{\infty} a_{n}$ is Lambert summable to $A$ and $n a_{n} \geq - C$ then $\sum_{n = 0}^{\infty} a_{n}$ converges to $A$ .

The Tauberian theorem was first proven by G. H. Hardy and John Edensor Littlewood but 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 around the Lambert Tauberian theorem was resolved by Norbert Wiener.

Examples

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

References

Jacob Korevaar (2004). Tauberian theory. A century of developments. Grundlehren der Mathematischen Wissenschaften. Vol. 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. Vol. 97. Cambridge: Cambridge Univ. Press. pp. 159–160. ISBN 978-0-521-84903-6.
Norbert Wiener (1932). "Tauberian theorems". Ann. of Math. 33 (1). The Annals of Mathematics, Vol. 33, No. 1: 1–100. doi:10.2307/1968102. JSTOR 1968102.

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Lambert summation

Definition

Abelian and Tauberian theorem

Examples

See also

References