Dirichlet series
التعريفات والمعاني
== English ==
=== Alternative forms ===
Dirichlet's series
=== Etymology ===
Named after German mathematician Peter Gustav Lejeune Dirichlet (1805–1859).
=== Noun ===
Dirichlet series (countable and uncountable, plural Dirichlet series)
(number theory) Any infinite series of the form
∑
n
=
1
∞
a
n
n
s
{\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}}
, where
s
{\displaystyle s}
and each
a
n
{\displaystyle a_{n}}
are complex numbers.
2014, Marius Overholt, A Course in Analytic Number Theory, American Mathematical Society, page 157,
The sum
A
(
s
)
=
∑
n
=
1
∞
a
n
n
−
s
{\displaystyle A(s)=\sum _{n=1}^{\infty }a_{n}n^{-s}}
of a convergent Dirichlet series is a holomorphic (single-valued analytic) function in the half plane
σ
>
σ
c
(
A
)
{\displaystyle \sigma >\sigma _{c}(A)}
, and the terms of the Dirichlet series are holomorphic in the whole complex plane, and the series converges uniformly on every compact subset of
σ
>
σ
c
(
A
)
{\displaystyle \sigma >\sigma _{c}(A)}
by Proposition 3.3.
==== Usage notes ====
In the above form, sometimes called the ordinary Dirichlet series.
Setting
a
n
=
1
{\displaystyle a_{n}=1}
yields
∑
n
=
1
∞
1
n
s
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{s}}}}
, which is the formula of the Riemann zeta function.
When rendered as
∑
n
=
1
∞
a
n
e
−
λ
n
s
{\displaystyle \sum _{n=1}^{\infty }a_{n}e^{-\lambda _{n}s}}
(for some sequence
{
λ
n
}
{\displaystyle \{\lambda _{n}\}}
for which
{
|
λ
n
|
}
{\displaystyle \{\vert \lambda _{n}\vert \}}
increases monotonically), may be called the general Dirichlet series.
This reduces to the ordinary form if
λ
n
=
ln
n
{\displaystyle \lambda _{n}=\ln n}
.
It is in the general form that the series is most plainly seen to be a special case of the Laplace-Stieltjes transform.
==== Synonyms ====
(infinite series): general Dirichlet series, ordinary Dirichlet series
==== Related terms ====
Dirichlet L-series
==== Translations ====
=== See also ===
Dirichlet function (unrelated concept)
Dirichlet L-function
Riemann zeta function
=== Further reading ===
Riemann zeta function on Wikipedia.Wikipedia
Dirichlet series on Encyclopedia of Mathematics
Dirichlet L-Series on Wolfram MathWorld
Dirichlet Series on Wolfram MathWorld