Lambert summation

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

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

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

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.

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

• Lambert series
• Abel–Plana formula
• Abelian and tauberian theorems

• 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. The Annals of Mathematics, Vol. 33, No. 1. 33 (1): 1–100. doi:10.2307/1968102. JSTOR 1968102.