Abel's summation formula

Other concepts sometimes known by this name are summation by parts and Abel–Plana 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 ${\displaystyle (a_{n})_{n=0}^{\infty }}$ be a sequence of real or complex numbers. Define the partial sum function ${\displaystyle A}$ by

${\displaystyle A(t)=\sum _{0\leq n\leq t}a_{n}}$

for any real number ${\displaystyle t}$. Fix real numbers ${\displaystyle x<y}$, and let ${\displaystyle \phi }$ be a continuously differentiable function on ${\displaystyle [x,y]}$. Then:

${\displaystyle \sum _{x<n\leq y}a_{n}\phi (n)=A(y)\phi (y)-A(x)\phi (x)-\int _{x}^{y}A(u)\phi '(u)\,du.}$

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

Variations

Taking the left endpoint to be ${\displaystyle -1}$ gives the formula

${\displaystyle \sum _{0\leq n\leq x}a_{n}\phi (n)=A(x)\phi (x)-\int _{0}^{x}A(u)\phi '(u)\,du.}$

If the sequence ${\displaystyle (a_{n})}$ is indexed starting at ${\displaystyle n=1}$, then we may formally define ${\displaystyle a_{0}=0}$. The previous formula becomes

${\displaystyle \sum _{1\leq n\leq x}a_{n}\phi (n)=A(x)\phi (x)-\int _{1}^{x}A(u)\phi '(u)\,du.}$

A common way to apply Abel's summation formula is to take the limit of one of these formulas as ${\displaystyle x\to \infty }$. The resulting formulas are

${\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }a_{n}\phi (n)&=\lim _{x\to \infty }{\bigl (}A(x)\phi (x){\bigr )}-\int _{0}^{\infty }A(u)\phi '(u)\,du,\\\sum _{n=1}^{\infty }a_{n}\phi (n)&=\lim _{x\to \infty }{\bigl (}A(x)\phi (x){\bigr )}-\int _{1}^{\infty }A(u)\phi '(u)\,du.\end{aligned}}}$

These equations hold whenever both limits on the right-hand side exist and are finite.

A particularly useful case is the sequence ${\displaystyle a_{n}=1}$ for all ${\displaystyle n\geq 0}$. In this case, ${\displaystyle A(x)=\lfloor x+1\rfloor }$. For this sequence, Abel's summation formula simplifies to

${\displaystyle \sum _{0\leq n\leq x}\phi (n)=\lfloor x+1\rfloor \phi (x)-\int _{0}^{x}\lfloor u+1\rfloor \phi '(u)\,du.}$

Similarly, for the sequence ${\displaystyle a_{0}=0}$ and ${\displaystyle a_{n}=1}$ for all ${\displaystyle n\geq 1}$, the formula becomes

${\displaystyle \sum _{1\leq n\leq x}\phi (n)=\lfloor x\rfloor \phi (x)-\int _{1}^{x}\lfloor u\rfloor \phi '(u)\,du.}$

Upon taking the limit as ${\displaystyle x\to \infty }$, we find

${\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }\phi (n)&=\lim _{x\to \infty }{\bigl (}\lfloor x+1\rfloor \phi (x){\bigr )}-\int _{0}^{\infty }\lfloor u+1\rfloor \phi '(u)\,du,\\\sum _{n=1}^{\infty }\phi (n)&=\lim _{x\to \infty }{\bigl (}\lfloor x\rfloor \phi (x){\bigr )}-\int _{1}^{\infty }\lfloor u\rfloor \phi '(u)\,du,\end{aligned}}}$

assuming that both terms on the right-hand side exist and are finite.

Abel's summation formula can be generalized to the case where ${\displaystyle \phi }$ is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral:

${\displaystyle \sum _{x<n\leq y}a_{n}\phi (n)=A(y)\phi (y)-A(x)\phi (x)-\int _{x}^{y}A(u)\,d\phi (u).}$

By taking ${\displaystyle \phi }$ to be the partial sum function associated to some sequence, this leads to the summation by parts formula.

Examples

Harmonic numbers

If ${\displaystyle a_{n}=1}$ for ${\displaystyle n\geq 1}$ and ${\displaystyle \phi (x)=1/x,}$ then ${\displaystyle A(x)=\lfloor x\rfloor }$ and the formula yields

${\displaystyle \sum _{n=1}^{\lfloor x\rfloor }{\frac {1}{n}}={\frac {\lfloor x\rfloor }{x}}+\int _{1}^{x}{\frac {\lfloor u\rfloor }{u^{2}}}\,du.}$

The left-hand side is the harmonic number ${\displaystyle H_{\lfloor x\rfloor }}$.

Representation of Riemann's zeta function

Fix a complex number ${\displaystyle s}$. If ${\displaystyle a_{n}=1}$ for ${\displaystyle n\geq 1}$ and ${\displaystyle \phi (x)=x^{-s},}$ then ${\displaystyle A(x)=\lfloor x\rfloor }$ and the formula becomes

${\displaystyle \sum _{n=1}^{\lfloor x\rfloor }{\frac {1}{n^{s}}}={\frac {\lfloor x\rfloor }{x^{s}}}+s\int _{1}^{x}{\frac {\lfloor u\rfloor }{u^{1+s}}}\,du.}$

If ${\displaystyle \Re (s)>1}$, then the limit as ${\displaystyle x\to \infty }$ exists and yields the formula

${\displaystyle \zeta (s)=s\int _{1}^{\infty }{\frac {\lfloor u\rfloor }{u^{1+s}}}\,du.}$

where ${\displaystyle \zeta (s)}$ is the Riemann zeta function. This may be used to derive Dirichlet's theorem that ${\displaystyle \zeta (s)}$ 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 ${\displaystyle a_{n}=\mu (n)}$ is the Möbius function and ${\displaystyle \phi (x)=x^{-s}}$, then ${\displaystyle A(x)=M(x)=\sum _{n\leq x}\mu (n)}$ is Mertens function and

${\displaystyle {\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}=s\int _{1}^{\infty }{\frac {M(u)}{u^{1+s}}}\,du.}$

This formula holds for ${\displaystyle \Re (s)>1}$.

See also

• Summation by parts
• Integration by parts

References

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