# Abel's summation formula

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 , and let $\phi$ be a continuously differentiable function on $[x,y]$ . Then:

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

$\sum _{0\leq n\leq x}\phi (n)=\lfloor x+1\rfloor \phi (x)-\int _{0}^{x}\lfloor u+1\rfloor \phi '(u)\,du.$ Similarly, for the sequence $a_{0}=0$ and $a_{n}=1$ for all $n\geq 1$ , the formula becomes

$\sum _{1\leq n\leq x}\phi (n)=\lfloor x\rfloor \phi (x)-\int _{1}^{x}\lfloor u\rfloor \phi '(u)\,du.$ Upon taking the limit as $x\to \infty$ , we find

{\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 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 $\Re (s)>1$ .