Lambert summation

Define the Lambert kernel by $L(x)=\backslash log(1/x)\backslash frac\{x\}\{1-x\}$ with $L(1)=1$. Note that $L(x^n)>0$ is decreasing as a function of $n$ when

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

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

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

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

• {{cite book | author=Jacob Korevaar | title=Tauberian theory. A century of developments | series=Grundlehren der Mathematischen Wissenschaften | volume=329 | publisher=Springer-Verlag | year=2004 | isbn=3-540-21058-X | pages=18 }}
• {{cite book | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) |author2=Robert C. Vaughan |authorlink2=Robert Charles Vaughan (mathematician) | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=978-0-521-84903-6 | pages=159–160 | publisher=Cambridge Univ. Press | location=Cambridge}}
• {{cite journal | author=Norbert Wiener | authorlink=Norbert Wiener | title=Tauberian theorems | journal=Ann. of Math. | year=1932 | volume=33 | pages=1–100 | doi=10.2307/1968102 | issue=1 | publisher=The Annals of Mathematics, Vol. 33, No. 1 | jstor=1968102 }}

Category:Series (mathematics)

Category:Summability methods

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