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.
Define the Lambert kernel by
( x ) = log ( 1
/
Note that
is decreasing as a function of
A sum
is Lambert summable to
lim
, written
Abelian theorem: If a series is convergent to
then it is Lambert summable to
Tauberian theorem: Suppose that
is Lambert summable to
Then it is Abel summable to
is Lambert summable to
converges to
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.
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