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(s, m) 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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a rationalzetaseries is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function...
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...
^{(r_{k}-1)}(b_{k}),} provided the series on the left converges. The digamma has a rationalzetaseries, given by the Taylor series at z = 1. This is ψ ( z + 1...
Taylor series for the Hurwitz zeta function. This series may be used to derive a number of rationalzetaseries. These non-converging series can be used...
+\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...
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...
{\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...
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...
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...
{\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...
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 ,...
obtainable through the types of resummation techniques used to obtain rationalzetaseries (Borwein et al. 2000). Recall the Barnes G-function, the Catalan's...
mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the...
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 (...
{\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...
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...
polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as...
f\colon X\to X} , the zeta-function is defined as the formal series ζ f ( t ) = exp ( ∑ n = 1 ∞ L ( f n ) t n n ) , {\displaystyle \zeta _{f}(t)=\exp \left(\sum...
many types of series with rational terms. In particular, it can be used to evaluate hypergeometric series at rational points. Given a series S ( a , b )...
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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