Abel's summation formula

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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]

  • Apostol, Tom (1976), Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag.