Integration by parts version of Abel's method for summation by parts
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.
Wikibooks has a book on the topic of: Analytic Number Theory/Useful summation formulas
Let
be a sequence of real or complex numbers. Define the partial sum function
by
![{\displaystyle A(t)=\sum _{0\leq n\leq t}a_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4656df57430d20768c569407515e0f949402eaa1)
for any real number
. Fix real numbers
, and let
be a continuously differentiable function on
. Then:
![{\displaystyle \sum _{x<n\leq y}a_{n}\phi (n)=A(y)\phi (y)-A(x)\phi (x)-\int _{x}^{y}A(u)\phi '(u)\,du.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2d8d9449b697eb78e27961d16c12190db481a3)
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
![{\displaystyle \sum _{0\leq n\leq x}a_{n}\phi (n)=A(x)\phi (x)-\int _{0}^{x}A(u)\phi '(u)\,du.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1941a4c70b9e49a717bbc702b898324d983254)
If the sequence
is indexed starting at
, then we may formally define
. The previous formula becomes
![{\displaystyle \sum _{1\leq n\leq x}a_{n}\phi (n)=A(x)\phi (x)-\int _{1}^{x}A(u)\phi '(u)\,du.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b83f3b793ceb7d0d01b244dfb66e2a39aebfa7c1)
A common way to apply Abel's summation formula is to take the limit of one of these formulas as
. The resulting formulas are
![{\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }a_{n}\phi (n)&=\lim _{x\to \infty }{\bigl (}A(x)\phi (x){\bigr )}-\int _{0}^{\infty }A(u)\phi '(u)\,du,\\\sum _{n=1}^{\infty }a_{n}\phi (n)&=\lim _{x\to \infty }{\bigl (}A(x)\phi (x){\bigr )}-\int _{1}^{\infty }A(u)\phi '(u)\,du.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbd06d0458f46b55b58dc85f4d118033ddfc4ed1)
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
![{\displaystyle \sum _{0\leq n\leq x}\phi (n)=\lfloor x+1\rfloor \phi (x)-\int _{0}^{x}\lfloor u+1\rfloor \phi '(u)\,du.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee851d1b0ee1e34c2a92b8f398f4f9ccf7b71610)
Similarly, for the sequence
and
for all
, the formula becomes
![{\displaystyle \sum _{1\leq n\leq x}\phi (n)=\lfloor x\rfloor \phi (x)-\int _{1}^{x}\lfloor u\rfloor \phi '(u)\,du.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f74bd9d1a3b42c1d617f34a3166f99cce424519)
Upon taking the limit as
, we find
![{\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }\phi (n)&=\lim _{x\to \infty }{\bigl (}\lfloor x+1\rfloor \phi (x){\bigr )}-\int _{0}^{\infty }\lfloor u+1\rfloor \phi '(u)\,du,\\\sum _{n=1}^{\infty }\phi (n)&=\lim _{x\to \infty }{\bigl (}\lfloor x\rfloor \phi (x){\bigr )}-\int _{1}^{\infty }\lfloor u\rfloor \phi '(u)\,du,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/164274453074f24a341d32b0cf98bbc37f4301a3)
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:
![{\displaystyle \sum _{x<n\leq y}a_{n}\phi (n)=A(y)\phi (y)-A(x)\phi (x)-\int _{x}^{y}A(u)\,d\phi (u).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d8999b207d45b7f31e780fbd324c078f2a8616c)
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
![{\displaystyle \sum _{n=1}^{\lfloor x\rfloor }{\frac {1}{n}}={\frac {\lfloor x\rfloor }{x}}+\int _{1}^{x}{\frac {\lfloor u\rfloor }{u^{2}}}\,du.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e966bf1bc81546a784feb2929505ebec06a95d4e)
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
![{\displaystyle \sum _{n=1}^{\lfloor x\rfloor }{\frac {1}{n^{s}}}={\frac {\lfloor x\rfloor }{x^{s}}}+s\int _{1}^{x}{\frac {\lfloor u\rfloor }{u^{1+s}}}\,du.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6034fce12c327e7a78053bed98adc1834a8ae8d)
If
, then the limit as
exists and yields the formula
![{\displaystyle \zeta (s)=s\int _{1}^{\infty }{\frac {\lfloor u\rfloor }{u^{1+s}}}\,du.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a0c1f0bd82e545d17a1a606343870436525b75e)
where
is the Riemann zeta function. 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
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
![{\displaystyle {\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}=s\int _{1}^{\infty }{\frac {M(u)}{u^{1+s}}}\,du.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1702bc225db7c6f60e8bdc1276465db2e52c9e16)
This formula holds for
.
See also
References