Abel's Summation Formula

In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in analytic number theory and the study of special functions to compute series.

Formula

Let $(a_n)_^\infty$ be a sequence of real or complex numbers. Define the partial sum function $A$ by
:$A(t) = \sum_ a_n$
for any real number $t$. Fix real numbers $x < y$, and let $\phi$ be a continuously differentiable function on $$, y/math>. Then:
:$\sum_ a_n\phi(n) = A(y)\phi(y) - A(x)\phi(x) - \int_x^y A(u)\phi'(u)\,du.$

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

Variations

Taking the left endpoint to be $-1$ gives the formula
:$\sum_ a_n\phi(n) = A(x)\phi(x) - \int_0^x A(u)\phi'(u)\,du.$
If the sequence $(a_n)$ is indexed starting at $n = 1$, then we may formally define $a_0 = 0$. The previous formula becomes
:$\sum_ a_n\phi(n) = A(x)\phi(x) - \int_1^x A(u)\phi'(u)\,du.$
A common way to apply Abel's summation formula is to take the limit of one of these formulas as $x \to \infty$. The resulting formulas are
:$\begin \sum_^\infty a_n\phi(n) &= \lim_\bigl(A(x)\phi(x)\bigr) - \int_0^\infty A(u)\phi'(u)\,du, \\ \sum_^\infty a_n\phi(n) &= \lim_\bigl(A(x)\phi(x)\bigr) - \int_1^\infty A(u)\phi'(u)\,du. \end$
These equations hold whenever both limits on the right-hand side exist and are finite.

A particularly useful case is the sequence $a_n = 1$ for all $n \ge 0$. In this case, $A(x) = \lfloor x + 1 \rfloor$. For this sequence, Abel's summation formula simplifies to
:$\sum_ \phi(n) = \lfloor x + 1 \rfloor\phi(x) - \int_0^x \lfloor u + 1\rfloor \phi'(u)\,du.$
Similarly, for the sequence $a_0 = 0$ and $a_n = 1$ for all $n \ge 1$, the formula becomes
:$\sum_ \phi(n) = \lfloor x \rfloor\phi(x) - \int_1^x \lfloor u \rfloor \phi'(u)\,du.$
Upon taking the limit as $x \to \infty$, we find
:$\begin \sum_^\infty \phi(n) &= \lim_\bigl(\lfloor x + 1 \rfloor\phi(x)\bigr) - \int_0^\infty \lfloor u + 1\rfloor \phi'(u)\,du, \\ \sum_^\infty \phi(n) &= \lim_\bigl(\lfloor x \rfloor\phi(x)\bigr) - \int_1^\infty \lfloor u\rfloor \phi'(u)\,du, \end$
assuming that both terms on the right-hand side exist and are finite.

Abel's summation formula can be generalized to the case where $\phi$ is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral:
:$\sum_ a_n\phi(n) = A(y)\phi(y) - A(x)\phi(x) - \int_x^y A(u)\,d\phi(u).$
By taking $\phi$ to be the partial sum function associated to some sequence, this leads to the summation by parts formula.

Examples

Harmonic numbers

If $a_n = 1$ for $n \ge 1$ and $\phi(x) = 1/x,$ then $A(x) = \lfloor x \rfloor$ and the formula yields
:$\sum_^ \frac = \frac + \int_1^x \frac \,du.$
The left-hand side is the harmonic number $H_$.

Representation of Riemann's zeta function

Fix a complex number $s$. If $a_n = 1$ for $n \ge 1$ and $\phi(x) = x^,$ then $A(x) = \lfloor x \rfloor$ and the formula becomes
:$\sum_^ \frac = \frac + s\int_1^x \frac\,du.$
If $\Re(s) > 1$, then the limit as $x \to \infty$ exists and yields the formula
:$\zeta(s) = s\int_1^\infty \frac\,du.$
This may be used to derive Dirichlet's theorem that $\zeta(s)$ has a simple pole with residue 1 at .

Reciprocal of Riemann zeta function

The technique of the previous example may also be applied to other Dirichlet series. If $a_n = \mu(n)$ is the Möbius function and $\phi(x) = x^$, then $A(x) = M(x) = \sum_ \mu(n)$ is Mertens function and
:$\frac = \sum_^\infty \frac = s\int_1^\infty \frac\,du.$
This formula holds for $\Re(s) > 1$.

See also

*Summation by parts
*Integration by parts

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.
Number theory
Summability methods