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

.

Abelian and Tauberian theorem

Abelian theorem.

Tauberian theorem.

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

• , where is the Möbius function in the ordinary sense. This is equivalent to the prime number theorem.

• is von Mangoldt function is Euler's constant is the second Chebyshev function.

See also

• Lambert series

• Abel–Plana formula

• Abelian and tauberian theorems

References