Rational series zeta

In Mathematical, a rational series zeta is the representation of a arbitrary Real number in terms of a series made up of rational numbers and Fonction zeta of Riemann or Fonction zeta of Hurwitz. More precisely, for a real number given X , the ratonelle series zeta for X is given by

$x=\; \backslash \; sum\_\; \{n=2\}\; ^\; \backslash \; infty\; q\_n\; \backslash \; zeta\; (N,\; m)$

where $q\_n\; \backslash ,$ is a rational number, the value m remains fixed and $\backslash \; zeta\; (S,\; m)\; \backslash ,$ is the function zeta of Hurwitz. It is not difficult to show that any real number X can be developed in this manner. For m whole, one has

$x=\; \backslash \; sum\_\; \{n=2\}\; ^\; \backslash \; infty\; q\_n\; \backslash \; left\; \backslash \; sum\_\; \{k=1\}\; ^\; \{M-1\}\; k^\; \{-\; N\}\; \backslash \; right$

For m=2 , much of interesting numbers have a simple expression in the form of rational series zeta:

$1=\backslash sum\_\{n=2\}^\backslash infty\; \backslash left$

and

$1-\; \backslash \; gamma=\; \backslash \; sum\_\; \{n=2\}\; ^\; \backslash \; infty\; \backslash \; frac\; \{1\}\; \{N\}\; \backslash \; left$

where $\backslash \; gamma\; \backslash ,$ is the Constante of Euler-Mascheroni. There exists also a series for $\backslash \; pi\; \backslash ,$:

$\backslash \; log\; \backslash \; pi\; =\; \backslash \; sum\_\; \{n=2\}\; ^\; \backslash \; infty\; \backslash \; frac\; \{2\; (3/2)\; ^n-3\}\; \{N\}\; \backslash \; left$

and

$\backslash \; frac\; \{13\}\; \{30\}\; -\; \backslash \; frac\; \{\backslash \; pi\}\; \{8\}\; =\; \backslash \; sum\_\; \{n=1\}\; ^\; \backslash \; infty\; \backslash \; frac\; \{1\}\; \{4^\; \{2n\}\}\; \backslash \; left$

becomes notable because of its fast convergence. This last series results from the general identity

$\backslash \; sum\_\; \{n=1\}\; ^\; \backslash \; infty\; (-\; 1)\; ^\; \{N\}\; t^\; \{2n\}\; \backslash \; left\; =$

\ frac {t^2} {1+t^2} + \ frac {1 \ pi T} {2} - \ frac {\ pi T} {e^ {2 \ pi T} -1}

who can be transformed starting from the generating Fonction of the numbers of Bernoulli

$\backslash \; frac\; \{X\}\; \{e^x-1\}\; =\; \backslash \; sum\_\; \{n=0\}\; ^\; \backslash \; infty\; B\_n\; \backslash \; frac\; \{t^n\}\; \{N!\}$

Adamchik and Srivastava give a similar series

$\backslash \; sum\_\; \{n=1\}\; ^\; \backslash \; infty\; \backslash \; frac\; \{t^\; \{2n\}\}\; \{n^2\}\; \backslash \; zeta\; (2n)\; =$

\ log \ left (\ frac {\ pi T} {\ sin (\ pi T)}\ right)

Series connected to the function polygamma

A number of additional relations can be deduced starting from the Taylor series for the Fonction polygamma at the point Z =1, which is

$\backslash \; psi^\; \{(m)\}(z+1)\; =\; \backslash \; sum\_\; \{k=0\}\; ^\; \backslash \; infty$

(- 1) ^ {m+k+1} (m+k)! \; \ zeta (m+k+1) \; \ frac {z^k} {K!}. This converges for | Z |<1. A particular case is

$\backslash \; sum\_\; \{n=2\}\; ^\; \backslash \; infty\; t^n\; \backslash \; left\; =$

- T \ left + \ psi (1-t) - \ frac {T} {1-t} \ right

who remains valid for . Here, $\backslash \; psi\; \backslash ,$ is the Fonction digamma and $\backslash \; psi^\; \{(m)\}\backslash ,$ is the function polygamma. Many series implying the binomial coefficient can be derived:

$\backslash \; sum\_\; \{k=0\}\; ^\; \backslash \; infty\; \{k+\; \backslash \; nu+1\; \backslash \; choose\; K\}\; \backslash \; left$

\ zeta (\ nu+2)

where $\backslash \; naked\; \backslash ,$ is a Complex number. This is resulting from the development in series of the function zeta of Hurwitz

$\backslash \; zeta\; (S,\; x+y)\; =$

\ sum_ {k=0} ^ \ infty {s+k-1 \ choose s-1} (there) ^k \ zeta (s+k, X) taken with $y=-1$. Similar series can be obtained simply in algebra:

$\backslash \; sum\_\; \{k=0\}\; ^\; \backslash \; infty\; \{k+\; \backslash \; nu+1\; \backslash \; choose\; k+1\}\; \backslash \; left$

1

and

$\backslash \; sum\_\; \{k=0\}\; ^\; \backslash \; infty\; (-\; 1)\; ^k\; \{k+\; \backslash \; nu+1\; \backslash \; choose\; k+1\}\; \backslash \; left$

2^ {- (\ nu+1)}

and

$\backslash \; sum\_\; \{k=0\}\; ^\; \backslash \; infty\; (-\; 1)\; ^k\; \{k+\; \backslash \; nu+1\; \backslash \; choose\; k+2\}\; \backslash \; left$

\ naked \ left - 2^ {- \ naked}

and

$\backslash \; sum\_\; \{k=0\}\; ^\; \backslash \; infty\; (-\; 1)\; ^k\; \{k+\; \backslash \; nu+1\; \backslash \; choose\; K\}\; \backslash \; left$

\ zeta (\ nu+2) - 1 - 2^ {- (\ nu+2)}

For $n\; \backslash \; geq\; 0\; \backslash ,$ whole, the series

$S\_n\; =\; \backslash \; sum\_\; \{k=0\}\; ^\; \backslash \; infty\; \{k+n\; \backslash \; choose\; K\}\; \backslash \; left$

can be written like a finite series

$S\_n=\; (-\; 1)\; ^n\; \backslash \; left\; \backslash \; zeta\; (k+1)\; \backslash \; right$

This results from simple a recursive Relation $S\_n+S\_\; \{n+1\}\; =\; \backslash \; zeta\; (n+2)\; \backslash ,$. Then, the series

$T\_n\; =\; \backslash \; sum\_\; \{k=0\}\; ^\; \backslash \; infty\; \{k+n-1\; \backslash \; choose\; K\}\; \backslash \; left$

can be written in the form

$T\_n=\; (-\; 1)\; ^\; \{n+1\}\; \backslash \; left\; (-\; 1)\; ^k\; (n-k)\; \backslash \; zeta\; (k+1)\; \backslash \; right$

for $n\; \backslash \; geq\; 1$ whole. This results starting from the identity $T\_n+T\_\; \{n+1\}\; =\; S\_n\; \backslash ,$. This process can be applied recursively to obtain series finite for the general expressions of the form

$\backslash \; sum\_\; \{k=0\}\; ^\; \backslash \; infty\; \{k+n-m\; \backslash \; choose\; K\}\; \backslash \; left$

for the positive integers m .

Series of half-whole powers

Similar series can beings obtained by exploring the Fonction zeta of Hurwitz for the half-whole values. Thus, for example, one has

$\backslash \; sum\_\; \{k=0\}\; ^\; \backslash \; infty\; \backslash \; frac\; \{\backslash \; zeta\; (k+n+2)\; -\; 1\}\; \{2^k\}$

= \ left (2^ {n+2} - 1 \ right) \ zeta (n+2) - 1

Expressions in the form of series p

Adamchik and Srivastava give

$\backslash \; sum\_\; \{n=2\}\; ^\; \backslash \; infty\; n^m\; \backslash \; left\; =$

1 \, + \ sum_ {k=1} ^m K! \; S (m+1, k+1) \ zeta (k+1)

and

$\backslash \; sum\_\; \{n=2\}\; ^\; \backslash \; infty\; (-\; 1)\; ^n\; n^m\; \backslash \; left\; =$

-1 \, + \, \ frac {1-2^ {m+1}} {m+1} B_ {m+1} \, - \ sum_ {k=1} ^m (- 1) ^k K! \; S (m+1, k+1) \ zeta (k+1)

where $B\_k\; \backslash ,$ is the numbers of Bernoulli and $S\; (m,\; K)\; \backslash ,$ is the numbers of Stirling of second species.

Other series

Other constants have remarkable rational series zeta:

• Constant of Khinchin
• Constant of Apéry

References

• Jonathan Mr. Borwein, David Mr. Bradley, Richard E. Crandall, on Computational Strategies for the Riemann Zeta Function

• Victor S. Adamchik and H. Mr. Srivastava, Sum series off the zeta and related functions

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