Abel's summation formula

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Another concept sometimes known by this name is summation by parts.

In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in number theory to compute series.

Identity[edit]

Let be a sequence of real or complex numbers and a function of class . Then

where

Indeed, this is integration by parts for a Riemann–Stieltjes integral.

More generally, we have

Examples[edit]

Euler–Mascheroni constant[edit]

If and then and

which is a method to represent the Euler–Mascheroni constant.

Representation of Riemann's zeta function[edit]

If and then and

The formula holds for It may be used to derive Dirichlet's theorem, that is, has a simple pole with residue 1 in s = 1.

Reciprocal of Riemann zeta function[edit]

If is the Möbius function and then is Mertens function and

This formula holds for

See also[edit]

References[edit]