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 a_n \, be a sequence of real or complex numbers and \phi (x) \, a function of class \mathcal{C}^1 \,. Then

\sum_{1 \le n \le x} a_n \phi(n) = A(x)\phi(x) - \int_1^x A(u)\phi'(u) \, \mathrm{d}u \,

where

A(x):= \sum_{1 \le n \le x} a_n \,.

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

More generally, we have

\sum_{x< n\le y} a_n\phi(n) = A(y)\phi(y) - A(x)\phi(x) -\int_x^y A(u)\phi'(u)\,\mathrm{d}u \,.

Examples[edit]

Euler–Mascheroni constant[edit]

If a_n = 1 \, and \phi (x) = \frac{1}{x} \,, then A (x) = \lfloor x \rfloor \, and

 \sum_1^x \frac{1}{n} = \frac{\lfloor x \rfloor}{x} + \int_1^x \frac{\lfloor u \rfloor}{u^2} \, \mathrm{d}u

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

Representation of Riemann's zeta function[edit]

If a_n = 1 \, and \phi (x) = \frac{1}{x^s} \,, then A (x) = \lfloor x \rfloor \, and

 \sum_1^\infty \frac{1}{n^s} = s\int_1^\infty \frac{\lfloor u\rfloor}{u^{1+s}} \mathrm{d}u \,.

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

Reciprocal of Riemann zeta function[edit]

If a_n = \mu (n) \, is the Möbius function and \phi (x) = \frac{1}{x^s} \,, then A (x) = M(u) = \sum_{n \le x} \mu (x) \, is Mertens function and

 \sum_1^\infty \frac{\mu(n)}{n^s} = s \int_1^\infty \frac{M(u)}{u^{1+s}} \mathrm{d}u \,.

This formula holds for \Re(s) > 1 \,.

See also[edit]

References[edit]

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

External links[edit]

This article incorporates information from this version of the equivalent article on the French Wikipedia.