English Dictionary

Riemann zeta function

noun

(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.

The Riemann zeta function is a mathematical function that is important in number theory.

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.