English Dictionary

Riemann zeta function

名詞

不可算名詞 通常 可算名詞

(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.

日本語の意味

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。

Research in analytic number theory often revolves around the Riemann zeta function and its complex zeros.

解析数論の研究はしばしば、ディリクレ級数 ζ(s)=∑_{n=1}^∞ 1/n^s によって定義される関数（リーマンゼータ関数）とその複素零点を中心に展開します。

意味（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.

It is straightforward to show that the Riemann zeta function has zeros at the negative even integers and these are called the trivial zeros of the Riemann zeta function.

意味（2）

(countable) A usage of (a specified value of) the Riemann zeta function, such as in an equation.

When the eigenfunctions are characters, these eigenvalues are respectively polynomials, products of ordinary gamma functions, and products of Riemann zeta functions, with the appropriate complex variables.

Riemann zeta functions （ plural ）