Last Updated：2025/11/22
(number theory, analytic number theory, uncountable) The function ζ defined by the Dirichlet series 𝜁(s)=∑ₙ₌₁ ᪲1/(nˢ)=1/(1ˢ)+1/(2ˢ)+1/(3ˢ)+1/(4ˢ)+⋯, which is summable for points s in the complex half-plane with real part > 1; the analytic continuation of said function, being a holomorphic function defined on the complex numbers with pole at 1.
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Riemann zeta function
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Source Word
Riemann zeta function
Noun
uncountable
usually
countable
(number
theory,
analytic
number
theory,
uncountable)
The
function
ζ
defined
by
the
Dirichlet
series
𝜁(s)=∑ₙ₌₁ ᪲1/(nˢ)=1/(1ˢ)+1/(2ˢ)+1/(3ˢ)+1/(4ˢ)+⋯,
which
is
summable
for
points
s
in
the
complex
half-plane
with
real
part
>
1;
the
analytic
continuation
of
said
function,
being
a
holomorphic
function
defined
on
the
complex
numbers
with
pole
at
1.
Japanese Meaning
リーマンゼータ関数（ζ関数）は、ディリクレ級数 ζ(s)=∑ₙ₌₁ 1/(n^s) により定義され、実部が1より大きい複素数 s の半平面で収束する関数です。解析接続により、この関数は複素全体に拡張され、1に極を持つホロモルフィック関数（複素解析可能な関数）として扱われ、数論および解析数論で重要な役割を果たします。
Sense（1）
(number
theory,
analytic
number
theory,
uncountable)
The
function
ζ
defined
by
the
Dirichlet
series
𝜁(s)=∑ₙ₌₁ ᪲1/(nˢ)=1/(1ˢ)+1/(2ˢ)+1/(3ˢ)+1/(4ˢ)+⋯,
which
is
summable
for
points
s
in
the
complex
half-plane
with
real
part
>
1;
the
analytic
continuation
of
said
function,
being
a
holomorphic
function
defined
on
the
complex
numbers
with
pole
at
1.
（ plural ）
