Riemann zeta function

English

Alternative forms

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Etymology

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Noun

1. Template:Lb The function ζ defined by the Dirichlet series $\zeta (s)={\sum }_{n=1}^{\mathrm{\infty }}\frac{1}{n^s}=\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\cdots$, 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. Template:C
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   • 2009, Arthur T. Benjamin, Ezra Brown (editors), Biscuits of Number Theory, Template:W, page 195,

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

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

     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

     $I_k(T)=\frac{1}{T}\underset{0}{\overset{\mathrm{\infty }}{\int }}{\left|\zeta (\frac{1}{2}+it)\right|}^{2k}dt$

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

2. Template:Lb 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, Template:W 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.

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Usage notes

• The Riemann zeta function and generalizations comprise an important subject of research in Template:W.
• There are numerous analogues to the Riemann zeta function: such a one may be referred to as a Template:M, 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 Template:Pedia)
• 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

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Hypernyms

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Translations

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See also

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Further reading

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• Zeta-function on Template:W
• Riemann Zeta Function on Template:W
  • Riemann Zeta Function zeta(2) on Template:W
• Xi-Function on Template:W