Sea
a
n
{\displaystyle a_{n}\,}
sucesión de números reales o complejos y
ϕ
(
x
)
{\displaystyle \phi (x)\,}
C
1
{\displaystyle {\mathcal {C}}^{1}\,}
fórmula de sumación de Abel es
∑
1
≤
n
≤
x
a
n
ϕ
(
n
)
=
A
(
x
)
ϕ
(
x
)
−
∫
1
x
A
(
u
)
ϕ
′
(
u
)
d
u
{\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\,}
dónde
A
(
x
)
:=
∑
1
≤
n
≤
x
a
n
.
{\displaystyle A(x):=\sum _{1\leq n\leq x}a_{n}\,.}
de hecho, esto es la integración por partes para una integral de Riemann–Stieltjes .
De forma más general, se tiene
∑
x
<
n
≤
y
a
n
ϕ
(
n
)
=
A
(
y
)
ϕ
(
y
)
−
A
(
x
)
ϕ
(
x
)
−
∫
x
y
A
(
u
)
ϕ
′
(
u
)
d
u
.
{\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)\,\mathrm {d} u\,.}
editar
Si
a
n
=
1
{\displaystyle a_{n}=1\,}
ϕ
(
x
)
=
1
x
,
{\displaystyle \phi (x)={\frac {1}{x}}\,,}
A
(
x
)
=
⌊
x
⌋
{\displaystyle A(x)=\lfloor x\rfloor \,}
∑
n
=
1
x
1
n
=
⌊
x
⌋
x
+
∫
1
x
⌊
u
⌋
u
2
d
u
{\displaystyle \sum _{n=1}^{x}{\frac {1}{n}}={\frac {\lfloor x\rfloor }{x}}+\int _{1}^{x}{\frac {\lfloor u\rfloor }{u^{2}}}\,\mathrm {d} u}
la cual es una manera de representar la constante de Euler–Mascheroni .
editar
Si
a
n
=
1
{\displaystyle a_{n}=1\,}
ϕ
(
x
)
=
1
x
s
,
{\displaystyle \phi (x)={\frac {1}{x^{s}}}\,,}
A
(
x
)
=
⌊
x
⌋
{\displaystyle A(x)=\lfloor x\rfloor \,}
ζ
(
s
)
=
∑
n
=
1
∞
1
n
s
=
s
∫
1
∞
⌊
u
⌋
u
1
+
s
d
u
.
{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=s\int _{1}^{\infty }{\frac {\lfloor u\rfloor }{u^{1+s}}}\mathrm {d} u\,.}
Esta fórmula es válida para todo
s
{\displaystyle s}
ℜ
(
s
)
>
1
.
{\displaystyle \Re (s)>1\,.}
ζ
(
s
)
{\displaystyle \zeta (s)\,}
polo simple con residuo 1 en
s
=
1
{\displaystyle s=1}
editar
Si
a
n
=
μ
(
n
)
{\displaystyle a_{n}=\mu (n)\,}
función de Möbius y
ϕ
(
x
)
=
1
x
s
,
{\displaystyle \phi (x)={\frac {1}{x^{s}}}\,,}
A
(
x
)
=
M
(
x
)
=
∑
n
≤
x
μ
(
n
)
{\displaystyle A(x)=M(x)=\sum _{n\leq x}\mu (n)\,}
función de Mertens y
1
ζ
(
s
)
=
∑
n
=
1
∞
μ
(
n
)
n
s
=
s
∫
1
∞
M
(
u
)
u
1
+
s
d
u
.
{\displaystyle {\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}=s\int _{1}^{\infty }{\frac {M(u)}{u^{1+s}}}\mathrm {d} u\,.}
Esta fórmula se cumple para
ℜ
(
s
)
>
1
.
{\displaystyle \Re (s)>1\,.}
Apostol, Tom (1976), Introduction to Analytic Number Theory , Undergraduate Texts in Mathematics, Springer-Verlag
Datos: Q1659643