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 and the study of special functions to compute series.

Formula[edit]

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[edit]

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[edit]

Harmonic numbers[edit]

If for and then and the formula yields

The left-hand side is the harmonic number .

Representation of Riemann's zeta function[edit]

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 s = 1.

Reciprocal of Riemann zeta function[edit]

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 .

See also[edit]

References[edit]

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