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Rational zeta series information


In mathematics, a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function or the Hurwitz zeta function. Specifically, given a real number x, the rational zeta series for x is given by

where qn is a rational number, the value m is held fixed, and ζ(sm) is the Hurwitz zeta function. It is not hard to show that any real number x can be expanded in this way.

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Rational zeta series

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a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function...

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Riemann zeta function

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The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined...

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Digamma function

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^{(r_{k}-1)}(b_{k}),} provided the series on the left converges. The digamma has a rational zeta series, given by the Taylor series at z = 1. This is ψ ( z + 1...

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Extended real number line

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+\infty } as the set of all sequences ( a n ) {\displaystyle (a_{n})} of rational numbers such that every M ∈ R {\displaystyle M\in \mathbb {R} } is associated...

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Polygamma function

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Taylor series for the Hurwitz zeta function. This series may be used to derive a number of rational zeta series. These non-converging series can be used...

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Hurwitz zeta function

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n = 0 ∞ 1 ( n + a ) s . {\displaystyle \zeta (s,a)=\sum _{n=0}^{\infty }{\frac {1}{(n+a)^{s}}}.} This series is absolutely convergent for the given values...

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Multiplicative function

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{\zeta (s)^{2}}{\zeta (2s)}}} More examples are shown in the article on Dirichlet series. An arithmetical function f is said to be a rational arithmetical...

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Dedekind zeta function

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obtained in the case where K is the field of rational numbers Q). It can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies...

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Riemann hypothesis

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In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers...

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Local zeta function

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In number theory, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as Z ( V ,...

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Basel problem

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{\displaystyle \zeta (2)} is detailed in expository fashion most notably in Havil's Gamma book which details many zeta function and logarithm-related series and integrals...

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Polylogarithm

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polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as...

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List of mathematical series

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{\displaystyle E_{n}} is an Euler number. ζ ( s ) {\displaystyle \zeta (s)} is the Riemann zeta function. Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function...

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Regular language

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0}s_{L}(n){\frac {z^{n}}{n}}}\right).} The zeta function of a regular language is not in general rational, but that of an arbitrary cyclic language is...

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Clausen function

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obtainable through the types of resummation techniques used to obtain rational zeta series (Borwein et al. 2000). Recall the Barnes G-function, the Catalan's...

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List of mathematical functions

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Chebyshev polynomials Synchrotron function Riemann zeta function: A special case of Dirichlet series. Riemann Xi function Dirichlet eta function: An allied...

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Euler product

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{\zeta (s)^{2}}{\zeta (2s)}}.} Since for even values of s the Riemann zeta function ζ(s) has an analytic expression in terms of a rational multiple...

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Igusa zeta function

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In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, modulo p, p2, p3, and so on....

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Motivic zeta function

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algebraic geometry, the motivic zeta function of a smooth algebraic variety X {\displaystyle X} is the formal power series: Z ( X , t ) = ∑ n = 0 ∞ [ X (...

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Weil conjectures

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field with qk elements. Weil conjectured that such zeta functions for smooth varieties are rational functions, satisfy a certain functional equation, and...

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