Abel's summation formula

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

Identity

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

Euler–Mascheroni constant

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

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

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

• Summation by parts

References

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