# Abel's summation formula

In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in number theory to compute series.

## Identity

Let ${\displaystyle a_{n}\,}$ be a sequence of real or complex numbers and ${\displaystyle \phi (x)\,}$ a function of class ${\displaystyle {\mathcal {C}}^{1}\,}$. Then

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

where

${\displaystyle A(x):=\sum _{1\leq n\leq x}a_{n}\,.}$

Indeed, this is integration by parts for a Riemann–Stieltjes integral.

More generally, we have

${\displaystyle \sum _{x

## Examples

### Euler–Mascheroni constant

If ${\displaystyle a_{n}=1\,}$ and ${\displaystyle \phi (x)={\frac {1}{x}}\,,}$ then ${\displaystyle A(x)=\lfloor x\rfloor \,}$ and

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

which is a method to represent the Euler–Mascheroni constant.

### Representation of Riemann's zeta function

If ${\displaystyle a_{n}=1\,}$ and ${\displaystyle \phi (x)={\frac {1}{x^{s}}}\,,}$ then ${\displaystyle A(x)=\lfloor x\rfloor \,}$ and

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

The formula holds for ${\displaystyle \Re (s)>1\,.}$ It may be used to derive Dirichlet's theorem, that is, ${\displaystyle \zeta (s)\,}$ has a simple pole with residue 1 in s = 1.

### Reciprocal of Riemann zeta function

If ${\displaystyle a_{n}=\mu (n)\,}$ is the Möbius function and ${\displaystyle \phi (x)={\frac {1}{x^{s}}}\,,}$ then ${\displaystyle A(x)=M(x)=\sum _{n\leq x}\mu (n)\,}$ is Mertens function and

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

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