Lambert summation

Template:Short description 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 $\sum _{n=0}^{\mathrm{\infty }}{a}_n=A(\mathrm{L})$.

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 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

• $\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (n)}{n}=0(\mathrm{L})$, 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 ordinary 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

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