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 [math]\displaystyle{ L(x)=\log(1/x)\frac{x}{1-x} }[/math] with [math]\displaystyle{ L(1)=1 }[/math]. Note that [math]\displaystyle{ L(x^n)\gt 0 }[/math] is decreasing as a function of [math]\displaystyle{ n }[/math] when [math]\displaystyle{ 0\lt x\lt 1 }[/math]. A sum [math]\displaystyle{ \sum_{n=0}^\infty a_n }[/math] is Lambert summable to [math]\displaystyle{ A }[/math] if [math]\displaystyle{ \lim_{x\to 1^-}\sum_{n=0}^\infty a_n L(x^n)=A }[/math], written [math]\displaystyle{ \sum_{n=0}^\infty a_n=0\,\,(\mathrm{L}) }[/math].

Abelian and Tauberian theorem

Abelian theorem: If a series is convergent to [math]\displaystyle{ A }[/math] then it is Lambert summable to [math]\displaystyle{ A }[/math].

Tauberian theorem: Suppose that [math]\displaystyle{ \sum_{n=1}^\infty a_n }[/math] is Lambert summable to [math]\displaystyle{ A }[/math]. Then it is Abel summable to [math]\displaystyle{ A }[/math]. In particular, if [math]\displaystyle{ \sum_{n=0}^\infty a_n }[/math] is Lambert summable to [math]\displaystyle{ A }[/math] and [math]\displaystyle{ na_n\geq -C }[/math] then [math]\displaystyle{ \sum_{n=0}^\infty a_n }[/math] converges to [math]\displaystyle{ A }[/math].

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

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