Riemann zeta function

See also: Riemann zeta-function

English

Alternative forms

• Riemann zeta-function
• Riemann's zeta function

Etymology

Named after German mathematician Bernhard Riemann.

Noun

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

1. (number theory, analytic number theory, uncountable) The function ζ defined by the Dirichlet series $\textstyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+{\frac {1}{4^{s}}}+\cdots$, 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.
   • 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, Melvyn Bernard Nathanson, editors, Number Theory: New York Seminar 2003, Springer,, page 87:

     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.

   • 2008, Sanford L. Segal, Nine Introductions in Complex Analysis, Revised edition, Elsevier (North-Holland), page 397:

     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. Some mention of the Riemann zeta-function, and treatment of the prime number theorem as an asymptotic result have become a topic treated by writers of introductory texts in complex variables.

   • 2009, Arthur T. Benjamin, Ezra Brown (editors), Biscuits of Number Theory, Mathematical Association of America, page 195,

     The Riemann zeta function is the function $\textstyle \zeta (s)=\sum _{n=1}^{\infty }n^{-s}$ for $s$ a complex number whose real part is greater than 1. […] The historical moments include Euler's proof that there are infinitely many primes, in which he proves

     $\zeta (s)=\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p^{s}}}\right)^{-1}$

     as well as Riemann's statement of his hypothesis and several others. Beineke and Hughes then define the moment of the modulus of the Riemann zeta function by

     $I_{k}(T)={\frac {1}{T}}\int _{0}^{\infty }\left\vert \zeta \left({\frac {1}{2}}+it\right)\right\vert ^{2k}dt$

     and take us through the work of several mathematicians on properties of the second and fourth moments.

2. (countable) A usage of (a specified value of) the Riemann zeta function, such as in an equation.
   • 2005, Jay Jorgenson, Serge Lang, Pos_n(R) and Eisenstein Series, Springer, Lecture Notes in Mathematics 1868, page 134,

     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.

   • 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 Pseudo-Differential Operators, Springer,, page 68:

     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 regularized trace of the heat kernel, and we express the Riemann zeta function in terms of the eigenvalues of the Laplacian. Issues on the regions of convergence of the series defining the Riemann zeta functions are beyond the scope of this paper and hence omitted.

   • 2013, George E. Andrews, Bruce C. Berndt, Ramanujan's Lost Notebook: Part IV, Springer, page 367:

     Page 203 is an isolated page on which Ramanujan evaluates six quotients of either Riemann zeta functions or L-functions.

Usage notes

• The Riemann zeta function and generalizations comprise an important subject of research in analytic number theory.
• There are numerous analogues to the Riemann zeta function: such a one may be referred to as a zeta function, and some are given names of the form X zeta function. Many, but not all functions so named are generalizations of the Riemann zeta function. (See List of zeta functions on Wikipedia.Wikipedia )
• The Riemann zeta function is used in physics (such as in the theory of heat kernels and wave propagation), apparently for reasons more to do with the function's behavior than any approximation of reality.

Synonyms

• (analytic continuation of a function defined as the sum of a Dirichlet series): Euler–Riemann zeta function

Hypernyms

• (analytic continuation of a function defined as the sum of a Dirichlet series): analytic function, zeta function

Translations

analytic continuation of a function defined as the sum of a Dirichlet series

• Dutch: Riemann-zeta-functie
• German: riemannsche ζ-Funktion f
• Italian: funzione zeta di Riemann f
• Spanish: función zeta de Riemann f
• Turkish: Riemann zeta fonksiyonu

See also

• Dirichlet eta function
• Dirichlet series
• L-function
• L-series
• Riemann hypothesis
• zeta function

Further reading

• Particular values of Riemann zeta function on Wikipedia.Wikipedia
• Dirichlet series on Wikipedia.Wikipedia
• Riemann hypothesis on Wikipedia.Wikipedia
• Lehmer pair on Wikipedia.Wikipedia
• List of zeta functions on Wikipedia.Wikipedia
• Riemann Xi function on Wikipedia.Wikipedia
• Zeta-function on Encyclopedia of Mathematics
• Riemann Zeta Function on Wolfram MathWorld
  • Riemann Zeta Function zeta(2) on Wolfram MathWorld
• Xi-Function on Wolfram MathWorld