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Riemann zeta function

Riemann zeta function noun: 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.

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...
    • The Riemann zeta-function (which has no relation to the Weierstrass function of Chapter 8, and must not be confused with it) was originally of interest because of its connection with number theory. Since then it has served as the model for a proliferation of "zeta-functions" throughout mathematics....2008, Sanford L. Segal, Nine Introductions in Complex Analysis, Revised edition, Elsevier (North-Holland), page 397
    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
    • The ubiquity of the heat kernels is demonstrated in Sections 9 and 10 by constructing, respectively, the Green functions and the Riemann zeta functions for the Laplacians on compact Lie groups. The Riemann zeta function of the Laplacian on a compact Lie group is the Mellin transform of the...2007, M. W. Wong, “Weyl Transforms, Heat Kernels, Green Function and Riemann Zeta Functions on Compact Lie Groups”, in Joachim Toft, M. W. Wong, Hongmei Zhu, editors, Modern Trends in...

Examples of Riemann zeta function in a sentence

  • The ubiquity of the heat kernels is demonstrated in Sections 9 and 10 by constructing, respectively, the Green functions and the Riemann zeta functions for the Laplacians on compact Lie groups.2007, M. W. Wong, “Weyl Transforms, Heat Kernels, Green Function and Riemann Zeta Functions on Compact Lie Groups”, in Joachim Toft, M. W....
  • Page 203 is an isolated page on which Ramanujan evaluates six quotients of either Riemann zeta functions or L-functions.2013, George E. Andrews, Bruce C. Berndt, Ramanujan's Lost Notebook: Part IV, Springer, page 367:

Synonyms of Riemann zeta function

  • Euler–Riemann zeta function

Forms of Riemann zeta function

pluralRiemann zeta functions
alternative formRiemann zeta-function
alternative formRiemann's zeta function

Origin of Riemann zeta function

Named after German mathematician Bernhard Riemann.

Questions about Riemann zeta function

What does Riemann zeta function mean?
Riemann zeta function means 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.
How do you use Riemann zeta function in a sentence?
The ubiquity of the heat kernels is demonstrated in Sections 9 and 10 by constructing, respectively, the Green functions and the Riemann zeta functions for the Laplacians on compact Lie groups.
What part of speech is Riemann zeta function?
Riemann zeta function is a noun.
What is the plural of Riemann zeta function?
The plural of Riemann zeta function is Riemann zeta functions.

Words near Riemann zeta function in the dictionary

  1. Riemann sphere
  2. Riemann spheres
  3. Riemann sum
  4. Riemann surface
  5. Riemann surfaces
  6. Riemann zeta function
  7. Riemann zeta functions
  8. Riemann zeta-function
  9. Riemann-Lebesgue lemma
  10. Riemann-Roch theorem
  11. Riemann-Stieltjes integral

Source and license

Adapted from Wiktionary: Riemann zeta function (page history and authors). Text available under the Creative Commons Attribution-ShareAlike 4.0 license. Word.ws reformatted the entry and added links. Wiktionary data of 2026-09-02.

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