Abel's summation formula

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 and the study of special functions to compute series.

Formula

Let $(a_{n})_{n=0}^{\infty }$ be a sequence of real or complex numbers. Define the partial sum function $A$ by

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

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

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

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

Variations

Taking the left endpoint to be $-1$ gives the formula

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

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

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

A common application is to the sequence $a_{n}=1$ for all $n\geq 0$. In this case, $A(x)=\lfloor x+1\rfloor$. For this sequence, the formula simplifies to

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

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

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

In particular,

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

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

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

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

By taking $\phi$ 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 $n\geq 1$ and $\phi (x)=1/x,$ then $A(x)=\lfloor x\rfloor$ and the formula yields

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

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

Representation of Riemann's zeta function

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

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

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

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

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

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

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

See also

• Summation by parts

References

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