Lambert summation explained

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) (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 ^{inf ty} _{n =0} a _{n} is Lambert summable to A if \lim _{x\to 1 ^{-}} \sum ^{inf ty} _{n =0} a _{n} L(x ^{n)=A} , written \sum ^{inf ty} _{n =0} 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 ^{inf ty} _{n =1} a _{n} is Lambert summable to A . Then it is Abel summable to A . In particular, if \sum ^{inf ty} _{n =0} a _{n} is Lambert summable to A and na _{n\geq} - C then \sum ^{inf ty} _{n =0} 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 ^{inf ty} _{n =1} (\mu(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 (\mu(n))/(n) satisfies the Tauberian condition, therefore the Tauberian theorem implies \sum ^{inf ty} _{n =1} (\mu(n))/(n) =0 in the ordinary sense. This is equivalent to the prime number theorem.\sum ^{inf ty} _{n =1} (Λ (n) -1)/(n) =-2 \gamma( L ) where Λ 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

• Book: Jacob Korevaar . Tauberian theory. A century of developments . Grundlehren der Mathematischen Wissenschaften . 329 . . 2004 . 3-540-21058-X . 18 .
• Book: Hugh Montgomery (mathematician)

. Hugh L. Montgomery . Hugh Montgomery (mathematician) . Robert C. Vaughan . Robert Charles Vaughan (mathematician) . Multiplicative number theory I. Classical theory . Cambridge tracts in advanced mathematics . 97 . 2007 . 978-0-521-84903-6 . 159–160 . Cambridge Univ. Press . Cambridge.

• Norbert Wiener . Norbert Wiener . Tauberian theorems . Ann. of Math. . 1932 . 33 . 1–100 . 10.2307/1968102 . 1 . The Annals of Mathematics, Vol. 33, No. 1 . 1968102 .