Riemann zeta-function

Noun

Riemann zeta function (usually uncountable, plural Riemann zeta functions)

1. (usually) 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.2004, Paula B. Cohen, “Interactions between number theory and operator algebras in the study of the Riemann zeta function (d'apres Bost-Connes and Connes)”, in David Chudnovsky, Gregory Chudnovsky...

   Synonyms:

   Euler–Riemann zeta function

2. (countable, usually) 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.2005, Jay Jorgenson, Serge Lang, Posₙ(R) and Eisenstein Series, Springer, Lecture Notes in Mathematics 1868, page 134