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) (number theory, analytic number theory, uncountable) The function ζ defined by the Dirichlet series ζ ( s ) = ∑ n = 1 ∞ 1 n s = 1 1 s + 1 2 s + 1 3 s + 1 4 s + ⋯ {\displaystyle \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. 2009, Arthur T. Benjamin, Ezra Brown (editors), Biscuits of Number Theory, Mathematical Association of America, page 195, The Riemann zeta function is the function ζ ( s ) = ∑ n = 1 ∞ n − s {\displaystyle \textstyle \zeta (s)=\sum _{n=1}^{\infty }n^{-s}} for s {\displaystyle 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 ζ ( s ) = ∏ p prime ( 1 − 1 p s ) − 1 {\displaystyle \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 ) = 1 T ∫ 0 ∞ | ζ ( 1 2 + i t ) | 2 k d t {\displaystyle 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. (countable) A usage of (a specified value of) the Riemann zeta function, such as in an equation. 2005, Jay Jorgenson, Serge Lang, Posn(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. ==== 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 ==== === 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