“Abel's summation formula”的意思、由来-开放百科全书

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Abel's summation formula

释义

Formula
Variations
Examples
Harmonic numbers Representation of Riemann's zeta function Reciprocal of Riemann zeta function
See also
References

{{dablink|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 and the study of special functions to compute series.

Formula

Let be a sequence of real or complex numbers. Define the partial sum function by

for any real number . Fix real numbers , and let be a continuously differentiable function on . Then:

The formula is derived by applying integration by parts for a Riemann–Stieltjes integral to the functions and .

Variations

Taking the left endpoint to be gives the formula

If the sequence is indexed starting at , then we may formally define . The previous formula becomes

A common way to apply Abel's summation formula is to take the limit of one of these formulas as . The resulting formulas are

These equations hold whenever both limits on the right-hand side exist and are finite.

A particularly useful case is the sequence for all . In this case, . For this sequence, Abel's summation formula simplifies to

Similarly, for the sequence and for all , the formula becomes

Upon taking the limit as , we find

assuming that both terms on the right-hand side exist and are finite.

Abel's summation formula can be generalized to the case where is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral:

By taking to be the partial sum function associated to some sequence, this leads to the summation by parts formula.

Examples

Harmonic numbers

If for and then and the formula yields

The left-hand side is the harmonic number .

Representation of Riemann's zeta function

Fix a complex number . If for and then and the formula becomes

If , then the limit as exists and yields the formula

This may be used to derive Dirichlet's theorem that has a simple pole with residue 1 at {{math|s {{=}} 1}}.

Reciprocal of Riemann zeta function

The technique of the previous example may also be applied to other Dirichlet series. If is the Möbius function and , then is Mertens function and

This formula holds for .

References

{{citation|first=Tom|last=Apostol|authorlink=Tom Apostol|title=Introduction to Analytic Number Theory|publisher=Springer-Verlag|series=Undergraduate Texts in Mathematics|year=1976}}.

2 : Number theory|Summability methods

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